The arithmetic theory of loop groups

THE ARITHMETIC THEORY OF LOOP GROUPS

by HOWARD GARLAND

To my father, Max Garland, on the occasion of his seventieth birthday

O , I . Introduct ion.

We let R denote a commutat ive ring with unit, and 5r R the ring of all formal Laurent series

.~. a~t ~, ai~R ,

with coefficients in R. We let k denote a field, so that -~k is also a field. For any commutat ive ring R, with unit, we let R* denote the units in R. We let G~ k denote the group of ~cfk-rational points of a simply connected Chevalley group G, which we take to be simple. We let

cT: ~ * • ~ * +k*

denote the tame-symbol (see w 12, (I~.2o)) . Then as one knows from the work of Matsumoto, Moore, and Steinberg (see [i3] , [i8], and [2o]), there is, corresponding to c~ -1, a central extension

(O.X) I -+k*-+ GT--> G.~%-+ I,

of G.v k. Our first goal in this paper is to develop a representation theory for the group G~ (here, k may have arbitrary characteristic). These representations will be infinite-dimensional, with representation space defined over the field k. They are constructed using the representation theory of Kac-Moody Lie algebras (see [io], [I I], [In], [I5] , [8], w167 3, 6, and see w 3 of the present paper). Thus, let B=(B0~. j= 1 ..... t be a symmetrizable Caftan matrix and let g(B) be the corresponding Kac-Moody Lie algebra (see w 3, for the definitions). Roughly speaking, one constructs g(B) by mimicking the Serre presentation, with B in place of a classical Caftan matrix corresponding to a (finite-dimensional) semi-simple Lie algebra. In general, g(B) is in fact infinite-dimensional.

Partially supported by NSF Grant :~MCS76-xo435.

181

H O W A R D G A R L A N D

Returning to the group G~, one may then take the point of view that the Lie algebra of G~ is a g(B), for a suitable choice of B. More precisely, let g denote the Lie algebra of the Chevalley group G, and let A denote the set of all roots of g (relative to a Cartan subalgebra t)). Let al, . . . , a t ( t = r a n k g) denote the simple roots in g (relative to some fixed order). For ease of description, we have assumed g is in fact simple, and we let a 0 denote the highest root of A (relative to our fixed order). Let a t + x = - - a 0 , and let A denote the ( e + i ) • matrix A=(~'j)~.j=I ..... t+ l , where

A 0 = 2 ( a o ~)(a~, a~)-l, i , j = I , . . . , g + I . We refer to such an A as an arlene Cartan matrix, and note that an affine Cartan matrix is a symmetrizable Cartan matrix. Then the Lie algebra of G~ is a k-form of g(A).

For k a field of characteristic zero, Kac and Moody initiated a representation theory for g(B), B equal to a symmetrizable Cartan matrix (see w 6) (1). Moreover, in the special case when B==A is an affine Cartan matrix one can prove the existence of a Chevalley lattice for each of these Kae-Moody representations with dominant integral highest weight (see [7], w i I , and also w 6 of the present paper, which may be considered a continuation of [7])- This Chevalley lattice then allows one to develop a representation theory over an arbitrary field k, and in this way we obtain the desired representations for the group G~.

Moreover, our theory being Z-rational, then even when k = R or (3 (2), we can use the existence of a Chevalley lattice to develop a theory of arithmetic subgroups of G~, and prove the existence of a fundamental domain for F, using Siegel sets! (w167 17-21 ). We will expired on this description later on in the introduction.

I t should be mentioned at the outset that the present formulation of our theory substantially differs from the original version in an earlier manuscript, and owes a great deal to the insight of the referee. Thus, in the original version, given Gao k (k an arbitrary field), the algebra g(A) corresponding to Gaok, and a Kac-Moody representation n of g(A), corresponding to a dominant integral highest weight X (see w 6 for the definition), we constructed a central extension G~ of a quotient of G~ k- The referee explicitly computed the symbol of G2 and found it to be a power of the tame symbol! (see w 12, Theorem (12.24)). Incidentally, one can always use Gk x to construct a central extension EZ(G~k ) of G.~k, rather than of a quotient of G.~k, by taking a suitable fiber product.

The referee also gave an elegant cohomological interpretation of the Kac-Moody algebra g(A) (w167 1-3, below). Thus, let k be a field of characteristic zero, and let k[t, t -1] be the ring of finite Laurent polynomials

Q

Y~ ait ~, aiek, i=i0

(a) It was in this context that V. Kae proved his celebrated character formula (see [I I]). (2) Throughout this paper we let Z, Q , R, and C denote the rational integers, rational numbers, real

numbers, and complex numbers, respectively.

182


where io<i 1 are integers. Let "~-~k[t, t-1]| regarded as a Lie algebra over k (so ~" is an infinite-dimensional Lie algebra over k: we call it the loop algebra (of g)). Then the referee proved that H2(~, k), the 2nd Lie algebra cohomology of ~ with respect to trivial action on k, has dimension one. The referee's proof is given in w 2. The idea is to compute the "symbol" of an arbitrary central extension; i.e., to develop a Lie algebra analogue for the theory of Matsumoto, Moore, and Steinberg ([x3] , [I8] and [2o]). The Lie algebra version is simpler.

On the other hand Kac and Moody knew that g(A) was a central extension of (9 simple) with one-dimensional center. I t then follows easily that g(A) is the

universal covering ~ (see w i, Definition (I .6)) of ~ (see w 3, Theorem (3-I4))- Also, if one replaces ~ by ~ 'c=g | k, then (as observed by the referee) the a rgument of w 2 can be adopted to construct a universal covering ~cD ~ of ~c, but now with 2 ~ universal in an appropriate category (see w 5, Remarks (5. i I)). The universality of tic

then allows one, by Galois descent, to construct a central extension 1 of a semi-simple Lie algebra I over ~cfk, which splits over an unramified Galois extension .~a k, of ~ (see Theorem (C24) of Appendix I I I , and see the Remark following Theorem (C24)).

Next, consider the group of .o~k-rationat points L of a semi-simple, simply connected, linear algebraic group L which is defined over .ocak, and which splits over the unramified extension Lf k, of -~k. We let Gwk, denote the s rational points of L. Analogous to the situation for Lie algebras, one might ask whether one may construct from the central extension Gk T, of Ga,k,, a central extension l', of L, by Galois descent. Of course one would expect that such a construction could be effected from a suitable universal property of the tame symbol. Indeed, when k' is a finite field, one could probably construct l'. by using Moore's theorem on continuous K 2 of .o~ k, (see, e.g., Milnor [I4], Appendix). However, in Appendix I I I of this paper, we rather consider the case when char k z o, and then we take a different approach: In order to construct the group I',, we utilize the universality property of ~, and we utilize the Kac-Moody representations of g(A), which correspond to dominant integral highest weights (these representations extend from g to fi~). However, we must now allow to be semi-simple and then give a suitable definition for .~.

Now we had said that we develop a representation theory for the groups G~. Indeed, for each dominant integral highest weight ?~, we obtain such a representation, and we obtain a corresponding image G~ of G T (see Definition (7.2I) for the definition o f ^ )' Gk). We define an Iwahori subgroup J CGk x (see Definition (7.24))- Roughly speaking, the subgroup or corresponds to the pullback of an Iwahori subgroup of G.~, k (see [9], [5], and the remark at the end of w 7).

(Gk, J , N, S) (see Theorem (I 4 . IO)). In w167 r I, I3, I4, we construct a Tits system ^x Again, roughly speaking, this Tits system is the pullback of the Tits system of [9] ( though our N is smaller).

Gk, considered In w167 9, to, I2, we obtain explicit information about the group ^~ as a central extension of the Elk-rational points of a classical Chevalley group. In

183


^), particular, in w I 2 , we associate to Gk, a central extension Ex(G~,k) of Gsok, and compute the symbol of this central extension (see Theorem (12.24)). This symbol turns out to be a power of the tame symbol. I t follows easily from this, and from the definition of the group EX(G~k), that we obtain a homomorphism from G~ onto Gk,̂ x i.e., a representation of G~. In w 15 we compare the G~ as X varies.

Now let k = 11 or t3 (from w I6 on, with the exception of the appendices, we pretty much restrict to the case when k = I t or G). Let Vk x denote the representation space corresponding to the dominant integral highest weight X for g(A). Then V x has a Chevalley lattice Vz x (so in particular, Vz x C Vk x is a Z-submodule such that

x )~ x V k =k | and V k has a positive-definite, Hermitian inner product { , } which is coherent with Vz x in the sense that {vl, v~}eZ, for Vl, v~eVz x (see [7], w167 11, I2, and see w167 6, 9 of the present paper). We let J = Z when k = R, and we let J be the ring of integers in a Euclidean, imaginary quadratic field, when k = 13 (see Remark (v) following the statement of Theorem (19.3)). We set V x = J Q z V z x and let F (resp. K) be the subgroup of ^x Gk, consisting of all elements which leave V~ (resp. { , }) invariant. We take the point of view that K C ~x is the analogue of a maximal compact subgroup, and letting J play the role of a parabolic subgroup, we prove the

^x I~J , in w 16 (see Theorem (16.8) and existence of an Iwasawa decomposition Gk= Lemma (16.14)). We then use this Iwasawa decomposition to define the notion of a Siegel set (see Definition (19.2), and the definition of ~ ( a > o ) , preceeding Lemma (20. io)).

Now in w 3 we introduce the degree derivation D = D t + 1 of g(A). From w 6 we then know that D acts on each Vk x, and then in w 17 we define the automorphism e ~B, cek, of V~. I t is apparent from the definitions of e ~D and ^x G k, that e ~D normalizes ^ x Gk, cek. Then using our Siegel set, we construct a fundamental

^ k -- r D domain for F acting on Gke r > o (see Theorem (2o.i4)) . Our proof of Theorem (2o. I4) is modeled on the proof of Theorem (1.6), of [1]. However, in the infinite dimensional case it takes extra work to prove the existence of minima. Actually we pass from F to a subgroup F0, to construct the fundamental domain in Theorem (2o. I4). Even for F0, our fundamental domain is not exact. However, for F0, one obtains a sharp description of the self intersections when r is sufficiently large (see Theorem (21. I6) and its Corollary i - - these results are analogues of the Harish-Chandra finiteness theorem and of the theorem of "transformations at ~ " , in the classical theory of fundamental domains). Our proof of Theorem (21.16) is related to that of the corresponding result in [1] (see e.g., Theorem (4.4) of [1]). However, we must now contend with the existence of infinitely many Bruhat cells.

We may give an alternative point of view for the above theory of fundamental domains. Thus, for r ~ o , let F~=e-*DF0e*~ We may then take the point of view

that we are constructing a fundamental domain for F, acting on 6~ (see Corollary i to Theorem (2I. 16), and the paragraph preceding that Corollary). Finally, if

~ _ _ rD R -- rD --e ~r ,

184

T H E A R I T H M E T I C T H E O R Y O F L O O P G R O U P S 9

where ~0(r) is as in Corollary i to Theorem (,21.16), then by that corollary and the

paragraph preceding it, we have that ~Fo ^x ----Gk, and we obtain a sharp description

of the self intersections. Moreover, we may naturally regard ~ as a Siegel set constructed with Kr=e~Dt~e -rD, r > o , in place of F-,.

The paper is organized as follows: In w167 1- 3 we construct the universal covering o f ~ when g is simple; we show in w 3 (still assuming g simple) that g(A) is the universal covering of ~'. In w 4 we define the Chevalley form gz(A) of g(A), as in [7]. In w 5 we introduce completions of the algebras ~ and g(A). In w 6 we introduce the Kac-Moody representation theory of Kac-Moody algebras, and the Chevalley lattice Vz x of [7], in the g(A)-representation space V x corresponding to a dominant integral highest weight X. In w 7 we define the Chevalley group ~x (k a field of arbitrary characteristic, and X a dominant integral highest weight). We also intro-

~ X duce the Iwahori subgroup J C G k. In w 8 we study the adjoint representation of ^ X G k. In w167 9, IO, and 12, we study the groups Gk x as central extensions of classical Chevalley groups, and explicitly compute the symbol. In w167 i i , 13, and 14, we

A x construct the Tits system (Gk, J , N, S) (see Theorem (14. io)). The proof given here that ^x (Gk, J , N, S) is a Tits system, is different from the proof we gave originally, and follows a suggestion of J. Tits. Thus, in w i i , we apply the results of [4], and construct a donnde radicielle with valuation. When char k = o , a similar (but on the surface, slightly different) Tits system had been constructed by Marcuson in [12]. Also, with some restriction on k, such a Tits system had been constructed by Moody and Teo for the adjoint group (see [17] ). Both the construction of Marcuson and of Moody-Teo were valid in the context of general Kac-Moody Lie algebras, while our Tits system is valid only for Kac-Moody algebras corresponding to affine Cartan matrices. However, our construction complements that of Marcuson, in that we make no restriction on the field k, and it complements that of Moody and Teo, in that we make no restriction on k and work with nonadjoint groups.

In w 15 we study the relation among the ^x Gk, as X varies. Finally, in w167 17-2 i, we construct the fundamental domain for the ari thmetic group F0. In w 17 we prove the existence of min ima of certain matrix coefficients of Vk x (k = R or G) on F0 orbits. In w 21, we prove our theorem on self intersections (Theorem (21. I6)). We mention that as a consequence of Theorem (2I . I6) , P0 and Pr are not conjugate in ~x ( k = R or G) for r > o sufficiently large (see Corollary (3) to Theorem (21.16)).

I n Appendix I, we prove L e m m a (11.2). Also, L e m m a (A. I ) of Appendix I is used to prove (17 . i ) . In Appendix II , we consider the case when k is a finite field, and formulate a conjecture about the special representation of G.~k, relating this representation to a suitable V~ (this conjecture is an analogue to the known

theorem for finite Chevalley groups, relating the Steinberg representation to a highest weight module). (Added in proof. - - J. Arnon recently proved a slightly modified version of this conjecture.)

185

2

I O H O W A R D G A R L A N D

As it stands, our theory is a theory for Chevalley groups Ga% , and their Lie

algebras. In Appendix I I I we begin investigating how to extend our results to non-split groups. Thus, assume char k = o, and as before, let L be the group of Q-ra t ional points of a linear algebraic group which is semi-simple and defined over 4 . Assume further that L splits over an unramified Galois extension 4 ' of .Sf k. Let G~k, denote the group of -5fk,-rational points of L. Following a suggestion of the referee, we show (by universality!) that at the Lie algebra level, we can lift Galois automorphisms from the Lie algebra of Gao k to that of the central extension G~. We then use this Lie algebra result and representation theory, to obtain a similar result at the group level, and then, by Galois descent, construct a central extension i',

of L. Our method of construction of i',, here, is somewhat different from the method of construction proposed by the referee. As we mentioned earlier, one could probably also construct L when k :is a finite field by using Moore's theorem (see Milnor [i4] , Appendix).

I f k = R or 13, and if L is actually defined over k, then the referee has extended the Iwasawa decomposition of w I6 to L and to i'~. Thus the stage is set for beginning the extension of our reduction theory for arithmetic groups to the non-split case.

I t is now apparent that the restrictions on the field k vary in the course of the paper. In w167 i, 2, 3, the meaning o f k is always made explicit, and for the most part, we assume in these sections that char k = o. In w167 4, 5, 6, the meaning of k is made explicit. In w167 7, 8, and io-i5, no restriction is made on the field k (so k may have arbitrary characteristic). From w i6 through the final section, w 2 i, we assume k is R or C. In Appendix I, no restriction is made on k. In Appendix II , we assume k is finite. In Appendix I I I , we assume char k = o .

As we have mentioned, we are indebted to the referee for communicating the

universality results of w167 I, 2, and 3, the relation in w 12, between the central extension Gx k and the tame symbol, and the implications these last results have for extending our results to the non-split case. We are also indebted to J . Tits for a long and detailed correspondence and many suggestions. We extend to both, our hearty thanks. Also, we wish to thank D. Belli for her patient and painstaking efforts in typing the manuscript.

I. Genera l r e m a r k s on centra l e x t e n s i o n s o f Lie Algebras .

In this section we describe Lie algebra analogues of some of the results in Moore [I8], Chapter I. At the outset we make the notational convention that C, R, Q , and Z denote the fields of complex numbers, real numbers, rational numbers and the ring of rational integers, respectively. We let a denote a Lie algebra over a

field k. By a central extension of a by a Lie algebra b, we mean an exact sequence of

Lie algebras (over k):

(x.x) o-+b-+eL a--,o,

186

THE ARITHMETIC THEORY OF LOOP GROUPS I !

such that b is in the center of e. I f t 'r~t

( * . 2 ) o - + b ' ~ e -+ c t -~o

is a second central extension of a, then by a morphism from the central extension ( i . I) to the central extension (, .2), we mean a pair of Lie algebra homomorphisms (% ~b), q~ : e ~ e', ~ : b ~ b', such that the diagram

b > e

b' > e' ~

is commutative. We say a Lie algebra is perfect, if it is equal to its own commutator subalgebra.

Definition ( t .4 ) . - - We call the central extension (I. I) a covering of a, in case e is

perfect. In this case, we will also call e or ~ (or the pair (e, ~z)) a covering of a, and we will

say that e (or the pair (% ~)) covers a.

Remark. - - I f the Lie algebra a admits a covering, then a is perfect.

Lemma (x. 5). - - I f ( I . I) is a covering of a, then there is at most one morphism from the

central extension ( I . I) to a second central extension of a.

Proof. - - Assume (% +) and (q~', •') are morphisins from the central extension ( i . i ) to the central extension ( i .~) . Thus, in particular, we also have a commutative diagram (I .3) with (~', +') in place of (% @. For x, y e e , we consider

(~ - - ? ' ) ([x ,y]) = ? ( [ x , y ] ) -- q/(Ix, y]) =

---- [q0 (x) -- ?'(x), q0(y)] + [q~'(x), q~(y) -- q0'(y)]

O 7

where the last equality follows from the fact that ~(z) - -q / (z )eb ' , for all zEe, this last assertion following from our assumption that (I .3) is commutative both for (% +)

and for (?', +').

Definition ( i . 6). - - We say that a covering of a is universal, i f for every central exten-

sion of a, there is a unique morphism (in the sense of central extensions) from the covering to the

central extension.

Remark. - - In view of Lemma (I "5), it suffices, in order to verify that a given

covering is universal, to show that for every central extension of a, there exists a

morphism from the given covering to the central extension. Uniqueness then follows

automatically from Lemma (I. 5)-

187

I 2 H O W A R D G A R L A N D

The following proposition is an immediate consequence of Definition (I. 6):

Proposition (x.7). - - Any two universal coverings of a are isomorphic as central extensions.

Of course, by Lemma (I. 5), the isomorphism is unique. Now let V be a vector space over k, and let Z~(a, V) denote the vector space

over k, of all skew-symmetric, bilinear maps

f : a • a-+V,

such that

(x.8) f ([x ,y] , z)--f([x, z ] , y ) + f ( [ y , z], x ) = o x,y, z~a.

We let B2(a, V) denote the vector space of all f , as above, such that there exists a linear map g : a ~ V , with

f (x ,y ) =g([x,y]) , x, yea .

Then the Jacobi identity implies that B2(a, V)C Z~(a, V), and we set

2(a, v ) = z (a, V).

We call Z2(a, V) (resp. B2(a,V), resp. H2(a,V)) the space of 2-cocycles (resp. 2-coboundaries; resp. the second cohomology group of a) with respect to V (regarded as a trivial a-module). It is well known that H2(a, V) parametrizes (suitably defined) equivalence classes of central extensions

o - + g - + e--+ a - -+ o .

We briefly recall how one constructs such a central extension, given an element f in Z2(a, V). We let

a/= (l|

and if 4=(x , v), ~ = ( y , w)~a! (so x, yea , v, w~V) we define the bracket [4, ~]~a! by

( I .9) [4, ~]=([x ,y] , f (x ,y ) ) .

It follows directly from the cocycle identity ( i . 8) that this bracket satisfies the Jacobi identity, and so with this bracket, a t is a Lie algebra, and the exact sequence

Yo o - ~ V - ~ a/-+ a-+ o,

where : is the inclusion, and ~ the projection from a I onto its first factor, is a central extension of a.

In the next section we shall give an explicit construction of the universal covering,

in a special case. We have already noted that if the Lie algebra a admits a covering, then a is perfect. Conversely, we have:

188

T H E A R I T H M E T I C T H E O R Y OF L O O P G R O U P S 13

Lemma (x. xo). - - I f a is perfect, then a has a universal covering.

Proof. - - We let W ' = A2a denote the second exterior power of a, and we let I C W' denote the subspace spanned by all elements

[ x , y ] A z - - [ x , z ] A y + [ y , Z]AX, x ,y , z ea .

We then set W = W ' / I , and let 0~(x,y)eW denote the image of xAyEW' . From the definition of I, one immediately sees that :r satisfies the cocycle identity (I .8); i.e., ~ Z ~ ( a , W).

We let

(x. xx ) o - + W - + as-+ a -+o

denote the corresponding central extension.

I f f e Z 2 ( a , V) and

( , . ,2) o ~ V - + al-~ a-+o

is another central extension of a, we consider the map ~b':W--~V defined by t~ '(e(x,y))~-- f(x ,y) , x, yEa . We then define ? ' : a:,-+a! by ?'(x, u)=(x, +(u)), xea ,

u ~ W .

It is easily checked that (?', +') is a morphism from the central extension (I. I I) to the central extension (i .I2).

Now let 3 = [ % , as] denote the comnmtator subalgebra of a~. Since a is perfect, we have

a + W = a~,

and hence ~ = [a,, a~] = [~, ~].

Thus, if c=-Wnfi , then the central extension

(I. I3) 0 -->" { -->" a -->" a - + 0

is a covering of a, and if

= ~ ' restricted to r

? = ? ' restricted to a

then (?, +) is a morphism from this central extension to the central extension ( i . 12). This proves that the central extension ( i . 13) is a universal covering of a. �9

The remainder of this section will not be needed in the sequel, but is included to complete the analogy with some of the results in Moore [I8]. Thus:

Definition ( I . i 4 ) . - - We will say a Lie algebra a is simply connected, in case it is

true that for every central extension 7~

0--~ b --~ e - ~ a - ~ o ,

of a, there is a unique homomorphism ? : a - + e , such that r:o?=identity.

189

x4 H O W A R D G A R L A N D

Definition (i.iS), - - - I f II is a Lie algebra and

7~

o - ~ b ~ e - §

a covering of a, with e simply connected, then we call this a simply connected covering of a.

Theorem (I . 16). - - A covering of a is universal i f and only i f it is simply connected.

Proof. - - Let ^ ~

( I . I 7 ) O--> C --> a --> a--> O

be a simply connected covering of a, and let us consider a central extension (I. I) of a. We may "lift" this central extension to a as follows: We let e(a) C e • denote the subalgebra of all ( x , y ) ee • such that r~(x)=~(y) . The projection of e • onto the second factor, induces a surjective homomorphism

D ^ e(a)

and it is easily checked tlhat kernel p is contained in the center of e(a). Thus, since is simply connected, there is a unique homomorphism ~:~--->e(a), such that

q~op=identity. We let X : ~--->e, denote the composition of ? with the projection of e(a) onto the first factor of e • a. It is easily checked that X(c) C b, and hence, if we let ~ denote the restriction of X to c, then (X, ~) is a morphism from ( i . 17) to (I . I); i.e., the covering ( i . i7) is universal (since the morphism is automatically unique,

thanks to Lemma (1.5).

Conversely, assume (I. 17) is a universal covering of a. Let

e ~ v ~

( I . I 8 ) 0 --+ b --+ e --->" a ---> 0

be a central extension. Since ~ is perfect, Lemma ( i .5) implies there is at most one morphism from the covering

0 ---> 0 ---> fl ---> fl ---> 0

to the central extension (I. I8), and hence, to prove (I. 17) is a simply connected covering, it suffices to prove (I. I8) is a split exact sequence. Again using the assumption that a is perfect, and arguing as in the proof of Lemma (I. lO), we can show that ~ 'o=[~, 'g] is perfect. But if bo=~'0nb, then the exact sequence ( I . I8 ) will split if the exact sequence

o ~ b 0 ~ e o ~ a ~ o

is split; i.e., we can assume the exact sequence ( I . i 8 ) is a covering. We now

consider the diagram

e -->fl--> a-->O,

190

THE ARITHMETIC THEORY OF LOOP GROUPS 15

and we set y = ~o~. We wish to show that kernel y is conta ined in the center o f ~ .

But if xEkerne ly , then v (x )~cen te r~ . Hence

(I. I9) [x,y] ~kernel v, y e Z .

Hence Ix, [Y l ,Y~] ]= ~ y l , y 2 e Z ,

by the J a c o b i identi ty, ( i . I9), and the fact that ke rne lvCcen te r ' g . But then x e c e n t e r e ~, since we are assuming Z is perfect. Thus

(x. 20) o -+kerne l y_+~ v-+ a-+ o

is a central extension. Since we are assuming ( i . i 7 ) is universal, there is a morphism (% +) f rom (I . I7) to (I .20); i.e., we have a commuta t ive d iagram

o > C > 8

o -+ kernel y ~ ~'

;, [1 > 0

A

We will be done, if we show that ,; o q~ is the identi ty.

But by the commuta t iv i ty of the above d iagram

and hence for xe~,

r

I f Xl, X~e~, then

g([Xl, X2] ) = 'V o V( [Xl, X2] ) - [Xl, X2] = [g(Xl), 'q o V(X2)] ~- [Xl, g(x2) ] = O,

since g takes values in r But then g is identically zero, since ~ is perfect. Hence ,; o qo = identity. �9

Remarks. - - (i) I f a is s imply connected, then a is perfect. Indeed, if a is not

perfect, there is a non-trivial l inear ma p f : a/[a, a] --+k, and hence, if we consider the trivial central extension

o ~ k - + a |

with k - + a | denot ing the inclusion, and a | the projection, then, using f , there are two splitting homomorphisms, and this nonuniqueness contradicts the simple connect ivi ty of a. (ii) I f a has a s imply connected covering,

0 - ~ C - + ~ - + Ct --~ O,

then ~ covers every covering of a. Indeed, if

0 --~ b --+ e--~ a--+ o

191

i6 H O W A R D G A R L A N D

is a covering of a, we (:an, since a simply connected covering is universal, find a

morphism (q~, ~) from the first central extension:

0 > r > a > a > 0

0 > b > e

We then need only check q~ is surjective.

of the above d iagram implies e = eo + b ,

e = [e, e] = [e0, e0] c e0,

But if e0 = q~(~), and hence

then the commuta t iv i ty

and so e = e 0 and q~ is surjective.

2. An expl ic i t cons truc t ion for the un iversa l cover ing o f the loop Algebra.

We let g denote a split simple Lie algebra over a field k of characteristic zero.

Let kit, t -1] (with t an indeterminate) denote the ring of polynomials in t and t - j , with

coefficients in k. We let

~ = krt, t- 11%g,

we write ux for u| (u~:_k[t, t-l], xsg) and we define a Lie bracket on ~ by

(2. x) [uxl, vx~] = uv| [xl, x2],

where u, vek[t, t - l ] , Xl, x2eg, and [xl, x~] denotes the Lie bracket of Xl, x2 in g.

We regard ~' with the [)racket (2 . I ) as a Lie algebra over k (which of course is

infinite-dimensional). We call ~ the loop algebra of g. I t is easy to verify that ~" is

perfect (since g is) and hence ~' has a universal covering. In this section, we shall give

an explicit construction :for the universal covering

o ~ c--~-§

of ~, and in part icular , find dim k r = I. To determine the covering, it suffices to give

a corresponding cocycle zeZ~(~, k). We must then show that the central extension

constructed from % as in w i, is the universal covering of ~.

In order to define "5 we first define a k-bilinear pair ing

-r0 : k [t , t -1] • t -1] --~k.

Namely, we let

Zo(U , v ) : residue (udv),

for u, w k [ t , t- l] . For example,

= = l - - n i f n + r n = o

"%(t', t m) (o otherwise.

192

T H E A R I T H M E T I C T H E O R Y OF L O O P G R O U P S x 7

We have

(2 .2) o=~0(ulu~, .~)+~0( .3 .1 , u~) + ~o(U~U3, ~1), .1, .2, .~k[t , t-q.

We let ( , ) denote the Killing form on g, and we define -r by

"r(ux, vy)=--%(u, v)(x,y), u, vek[t, t-l], x, yeg.

A direct verification, using (2. I) and the invariance of the Killing form, shows that -r satisfies the cocycle identity ( i .8 ) , and hence -r~Z2(~',k). We let

denote the corresponding central extension, constructed as in w I. We shall now prove that this central extension is a universal cover of ~.

First note that if

o - + b - + e - + a-+o

is a central extension of the Lie algebra a, if x,y~ a and if x',y'~ e are inverse images of x,y, respectively (so ~ (x ' )=x , 7~(y')=y) then [x',y'] depends only on x andy , and

not on our choice of inverse images x',y'. We therefore denote [x',y'] by [x,y]'. Now, let g=~I2(k), the Lie algebra of all 2 X 2 trace zero matrices, with

coefficients in k. Let H, E+, E be the Chevalley basis of ~12(k):

o E o =(io +=(o ;), o) Let o ~ Z ~ g ' p ~ -+ g--~ o

be an arbitrary central extension of g-----~. For uek[t, t-~], we define elements u E ' ,

uH" in ~ ' by

, I [H, uE+]' uE~: = 4-

uH' = [E+, u E ] ' . t !

When U = l , we write E a , H ' for IE+, IH' , respectively.

( 2 . 3 ) p (uIt') = uH p(uE' ) = uE•

Thus for u, wk[t , t-a], we have

(2 .4) [uH', vH'] = [uH, vH]'~Z.

We set

( 2 . 5 )

Also, we have from (2.3) that for

, i "H' (2 .6) uE• = • [ , uE~].

{u, v}= [uH', vH'] = [uH, vH] ' eZ .

uek[t, t-l],

We note that

193

3


Thus, substituting the above expression for uE'., and using the Jacobi identity, we have

[vU', uE'. ] = • ( [ [vii', H'], uE'.] + [ii ', [vH', uE,. ]]) = • [vII',uEk]], by (2.4).

On the other hand, [vH', uE'.]=• mod Z, and hence the last expression in the above computation equals

• I/2 [H', • 2uvE'~].

But (2.6) implies that this equals • Thus, we have shown

(2.7) [vH', uE'.] = • 2uvE'., u, wk[t , t-'].

Next, we consider (for u, vek[t, t-l]):

], by (2.7) and the definition of vH',

t I =uvI-I +-~{u, v}, by the definition ofuvH', (2.3) and (2.5).

, I [ug+, vE']=uvH +-~{u, v}, u, vek[t, t-l].

Next, for u, vEk[t, t-x], we consider

[uE~: v E ' . ] = • ', uE'~], vE'.], by (2.7), ' 2

~([[ ' , = • H', vE~], ,E~] + [I~', [,•• rE'. ]]).

The first summand equals [• uE'.] by (2.7) , and the second summand equals zero, since clearly [uE'. , vE'.] eZ. Hence the last expression in the above computation

! t . equals [vE• uEL], i.e.,

[ .E ; , vE'] = - - [ . E ' , ~E'],

and hence, since char k = o,

(*.9) [ug,,, vE, .]=o, u, vek[t, t- l] .

194

Thus, we have shown

(2.8)

T H E A R I T H M E T I C T H E O R Y O F L O O P G R O U P S

We summarize the multiplication rules, (2.5), (2.7), (2.8), and (2.9) in

(2.xo) a) [uH', vH']----{u, v}, b) [vii', uE2] = • 2uvE',,

c) [uE+, vet]= uvH'+-~{u, v},

d) [uE',, vE',]=o, u, vek[t, t-l].

I9

Lemma (a .xz) . - - The pairing { , } defines a skew symmetric, k-bilinear mapping from k[t, t-X]• t -1] to k. Moreover, relations (2. io) define a Lie algebra structure on "~', i f and only i f { , } satisfies the relations

(2. x2) {u, . } = o {uv, w}+{wu, v}+{vw, u } = o , u, v, wek[t, t- l] .

Proof. - - The bilinearity and skew symmetry of{ , }, and the relation {u, u}=o, all follow from (2.5). To understand the second identity in (2.I2), we consider (u, v, w~k[t, t-x])

t t t t t [uH', [vE+, wE~]] + [wE:z, [uH', vE• + [vE+, [wE:F, uH']] =+{u, vw}• uv}• uw},

and hence the second relation of (2.12) is equivalent to the Jacobi identity for uH', vE'., wE~:. As the remaining Jacobi identities follow from (2.IO), we obtain the Lemma. �9

We now wish to study a skew symmetric, bilinear map

{ , }: k[t, t - ' ]• t - 1 ] ~ k ,

satisfying (2. I2). First, for uek[t, t-t], we let

M, : kit, t -t] -+ k[t, t-t],

denote the multiplication operator

M,(v)=uv, v~k[t, t-l].

We let f , : k [ t , t-1]-+k denote the k-linear transformation, defined by

f , (v)={u, v}, v~k[t, t-q.

We easily deduce from (2.12) that

(2. x3) L o M, + f , o M . = L . , u, vek[t, t-l].

Then, applying (2. I3) when U----l, we see that f x = o . But if u, u-Xek[t, t-x], and if' we set v = u - t in (2.I3) , we obtain

(2.x4) f l = o .f,-x= -- f ,o M,,-,, u, u- tek[ t , t-x].

195

oO H O W A R D G A R L A N D

Then, by (2. I3), (2. I4) , and a straightforward induction, we obtain

(2. x 5) ft" = f t o M,,,-~, ne Z.

Thus, for r, seZ, we have

{t', f }=f t r ( t ~) =ftoM,r by (2. I5) , = r{t, t "+'-1}.

But, interchanging r and s, and using skew symmetry of { , }, we then also have

{t r, s t '+s-1}.

Thus, we have:

Lemma (2. x 6 ). - - Let

{ , }:k[t , t -~]xk[t , t - ~ ] ~ k

be a skew symmetric, bilinear pairing, satisfying the relations (2.12).

(2. I7) {t', t~}= ~,,_ ~r{t, t - l} , or equivalently

(2. xS) {u, v } = - - R e s i d u e (udv){t, t - l } , u, vek[t, t- l] .

Then we have

Remark. - - In essence, Lemmas (2. II) and (2.16) imply that the central extension introduced earlier,

(2.x9) is a universal cover of ~, in the case when g--~ ~I~(k). Indeed, one need only check ~ is perfect. We shall check this in general later on (in any case it is easy). We also remark that this central extension is therefore non-trivial.

We shall now assume 9 split semi-simple over k, and we let

be an arbitrary central extension of ~. We consider the Lie algebra fl, and fix a splitting Cartan subalgebra [9 C ft. We let A denote the set of roots of g with respect to I), and we let

g = I)| II g ~

denote the root space decomposition Thus A C D*, the dual

space of D, and

(relative to D) of ~.

For each ~eA we pick an element E, e9 ~ and an element H~I ) , aeA, we have

[E~, E_~] = H ~

[H~, E•177

so that for all

196

(2 . fifO)

Now for conditions

THE ARITHMETIC THEORY OF LOOP GROUPS 2 I

(We will further normalize our choice of the E~'s in w 4). We let g(e) denote the three dimensional subalgebra spanned by the elements E • and H , , and we let ~'(~) be the subalgebra spanned by the elements ux, uek[t, t-t], xeg(~). We let

=

so that if we let p~ denote the restriction of p to ~"(00, we have a central extension

o

eeA, we have an isomorphism from ~I2(k) onto g(e) defined by the

E• ~ E •

H ~ H ~ .

We then have a corresponding isomorphism q), from ~I2(k) onto ~(a), defined by the conditions

uE• ~ uE• uek[t, t-l].

uH ~ uH~,

We may thus regard the central extension (2.20) as a central extension of s Then the elements uE' . , u t t ' defined earlier, correspond to the elements

( 2 . 2 x ) uE2 ~ = 4- _I [H~, uE• ~]' 2 uek[t, t-l],

uH~ = [E~, uE_ ~]'

in ~'(e), and if we set

{u, v}~ = [uH'~, vH'~], u, vek[t, t-l],

then we have that the relations (2. Io) are valid in ~ ' (e) , with H' , E• { , }, replaced by I-I~, E• { , }~, respectively. Of course, we may also regard these as being bracket relations in ~' .

We now wish to consider the brackets of elements (2.2I) (regarded as elements in ~') as ~ varies. First, we consider [uH'~, rE;], u, vek[t, t-l],

p (uH;) = uH~ (2.22) u, vek[t, t-l], o~, ~eA.

p ( v E ; ) =

Thus, we have

Now we have

~, ~e A. Now

[ i 1] [uH~, r E ; ] = uH~,~[H~, vE~ , by (2.2I) and (2.22)

= - [[uH~, H;] vE H , [uH' , ']] , �9

2

[uH'~,, H~] = o mod Z ! [uH~, vE~] = ~(H~)uvE~ mod Z,

197

0 2 H O W A R D G A R L A N D

and hence the last expression in the above computation, is equal to

-I [H~, ~(H~)uvE'~]=~(H~)uvE'~, by (2. Io) b). 2

Thus we have shown

(2.23) [uH;, eA, u, oek[t, t-l].

Next, consider (o~, ~eA, u, vek[t, t-l]):

[uH'~, vH~]=[[E'~, uE'~], vU~], by (2.2I) and (2.~2), t t t t = [[E',, vH~], uE_~] + [E~, [uE; ~, vH~]],

by the Jacobi identity,

= [-- a(H~)vE~,, uE; ~] + [E~,, e(H~)uvE;~], by (2. ~3),

= - - c~(H~) (uvH; I ( ) +~{v, u}~)-t-e(H~) uvH; ~-~{I, UV}e ,

by (2.IO), c),

----~(H~){v,u}~, by (2.I7). 2

On the other hand, we may interchange the roles of e and ~ in the above computation, and we thus obtain:

(2.24) [uH'=, v H ; ] - - e(H~){v, u}=-- 2

As a Corollary of (2.24), we obtain:

(2.25) If (0~, ~):t:o, then {u, v}=--

(where ( , ) is also used to denote the inner product on D* induced by the Killing form).

Next, for e, [~eA, H' a linear combination of the H'8, SeA, u, vek[t, t-l], we have from (2.~2) and (2.23):

(2.26) [[H', uE;], vE;] = e(p(H')) [uE;, rE;].

On the other hand, by the Jacobi identity, the left side of the above equality equals

(2.27)

Now

and

(2.28)

where N~, ~ek

198

[[H', rE;], uE'~] + [H', [uE;, rE;]].

[H', vE'~]=~(o(H'))vE'~, by (2.23),

[uE'~, vE'~]=-N~.~uvE'~+~ mod Z,

is defined by

[E~, E~] =N~,~E~+ ~.

T H E A R I T H M E T I C T H E O R Y OF L O O P G R O U P S

Hence, we have

[H', [uE;, vE;]] = (~ § ~)(p(H'))N~, ~uvE'~ + ~,

and hence (2.27) equals

~3(p (H')) [rE;, uE;] -t- (~ 4- ~)(p(H'))Na, ~uvE'~ + ~.

Comparing with the right side of (2.26), we obtain

by (2.23), (2.28),

(0~ § ~)(p(H')) [uE;, vE;] = (0~ -t- ~)(p(H'))N~, ~uvE" + ~.

then we may choose H' so that (~+~) (p(H ' ) )#o , and thus conclude

[uE;, vE'~]=N,,~uvE'~+~, ~ + ~ 4 o .

23

N 1 [E_ ~,, E~,] + N 2 [E~,, E_ ~,] = N 2 H~, -- Nt H~,,

199

If we define N i ( i = I , 2 ) in k by

[E~i , E_ ,] = HiE_ ~i' ( i '= I

then the last expression in (.) equals:

if i = 2 , and i ' = 2 if i = I ) ,

If ~4:--~,

(2.29)

We collect (2.23) , (2.24) , (2.29) , and (2. IO), c) and d) (with E'~, in place of t t t E l , H~ in place of H , and { , }~ in place of { , }) into:

(2.3 o) a) [uH;, vE'~]=~(H~)uvE'~,

b) [uH;, vH;]--- -e(H~){v, u}~--~(H~){u, v}~, 2 2

c) [uE', vE'~]=N~,~uvE'~+~, if e + ~ + o ,

I ~) [ut~', vE'_a]=uvH;+~{u, v}~, u, wk[t, t-'J, ~, ~ •

We will need

Lemma (o.3x). - - I f ~eA, if ~=~1§ ~t, ~ e A , and i f q , c2ek are such that

Ha = clH~1+ c2H~,

then for all uek[t, t-l] , we have t uH'~ = Cl UH;, + c2uH~ .

Pro@ - - We have (in g):

H~ = [E~, E ~], N- lEa = levi, E J ,

for some Nek*, the multiplicative group of k. We then have

(.) N-~H. = [[V~,, E~,], E_~] = [[E~,, E_ ~], E~,] + [E~,[E~,, E_a]].

24 H O W A R D G A R L A N D

and hence

(2.32) H~ = NN2H~, - - N N : H ~ .

On the other hand (see (2.21))

uH" = [E~, uE_ ~]', t !

uH~ = [E~i, uE_ ~i], i = 1, 2.

Hence N-luH'=--[[E~I, E~,], uE'_,], by (2.3 o) c), t v v v ! = [ [ ~ , , u E ,], %,] + 0~,[~,, ,E" j ]

----NI[uE'_~,, E;,]-i-N2[E~,, uE'_~,], by (2.3 o) c), = - - N l u H ; , + N ~ u H ; , , by (2 .2i ) ,

t

and hence uH" = NN~uH~I-- NNtuH~,,

and comparing with (2.32), we obtain the lemma. �9 We now fix an order on the roots A, and let ~t, - . . , ~t ( l = d i m ~)) denote the

corresponding set of simple roots. We define q~:~'-+~" by the conditions

~(uH=~)=uH'=,, uEk[t, t-a], i = : , . . . , t.

Then, q~ is a section of our central extension

o - + Z ~ ' L ~ - + o

(i.e., ~ : ~ ' - + ~ ' is a linear transformation such that p o ~ = i d e n t i t y ) , and, thanks to L e m m a (2.3I) , we have

(2 .33) ? ( u H ~ ) = u H ' , uek[t, t-'], a~A.

One knows that if

%(x,y)=,?([x,y])--[?(x) , ?(y)] , x,y~g,

then %eZ2(~, Z) and our given central extension is equivalent to the central extension (see (1.12))

( 2 . ~ ) o - + Z ~ , + ~ o

defined by %. Moreover, thanks to Lemmas (2.11) and (2.16) and also thanks to (2.25), we

have that if ~ is any long root, then, in view of (2.33), we may reformulate (2.3 o) as the simple assertion:

(2.35)

where

200

%(u| v| = -- Res(udv)(x,y)h~,

h; ={t , t-:'}~ (~' ~)

4

T H E A R I T H M E T I C T H E O R Y OF L O O P G R O U P S 2 5

From (2.35) we easily deduce:

Theorem (2.36). - - Let -ceZ~(g, k) be defined by:

�9 (u| v | x, yek[ t , t-l], u, w g .

Then the corresponding central extension N

A CO

(~"37) o--+k--+ g . ~ g ~ o

is the universal covering of ~.

Proof. - - Thanks to (2-35), the map

s) sh3

from ~ = ~ | to ~r = ~ | (where ~ , s~k) defines a morphism from the central extension (2.37) to the central extension (2.34)- Thus, to prove the theorem, we need only prove that ~ is perfect. Since ~ is perfect, it suffices to prove that [~ , ~] contains k. But (see ( i . 9)) this follows from

v(t| t - l | 4 4:o,

where we may take 0~eA to be any root. �9

3" Kac-Moody Lie algebras.

In this section we indicate a different method for constructing the central extension (2.37)- In fact, this alternate approach is part of the very general theory of a Kac-Moody Lie algebra associated with a symmetrizable Cartan matrix (for details the reader may consult the papers of Kac and Moody [io], [15] and [I6], and Garland-Lepowsky [8]).

Thus i f l is a positive integer, then we say that an ~• matrix B=(B~j)i,i=l ..... t is a symmetrizable Cartan matrix in case BoeZ for all i and j , B , = 2 for all i, B0<o whenever i 4:3", and finally, there exist positive rational numbers qt, . . . , qt such that

diag(ql, . . . , qt)B

is a symmetric matrix. (If B satisfies all but the last condition it is called a Cartan matrix, and the last condition is called the symmetrizability condition. In this paper we shall only be concerned with symmetrizable Cartan matrices.)

Now given a field k of characteristic zero, and given a symmetrizable Cartan matrix B, Kac and Moody have constructed a certain Lie algebra g(B) over k, and we now proceed to describe g(B) (also see Garland-Lepowsky, [8]).

One lets g t = gt(B) denote the Lie algebra on 3 t generators e~, f , h i ( i=~ , . . . , t) with relations

( s . x ) = o, [ e , , f j ] =

[h,,

201

4


for i, j = x, . . . , g, and with the relations

(3.2) (ad e,)-s0+a(ej) = o,

(adf)-BiJ+ l(f~) = o,

for i4:j, i , j = i , . . . , g . For each g-tuple ( n l , . . . , nt) of nonnegative (resp. nonpositive) integers not all

zero, define gl(nl, . . . , nt) to be the subspace of gl spanned by the elements

[eq, [e,~, . . . , [ei,_l , %] . . . ] ] ,

(resp. [ fq ' [ f2 ' "" "' [f~,-l'f~, ] "" "]])'

where e~ (resp. f~) occurs lnjl times. Also, define g l ( O , . . . , o)=I)l(B), the linear span of hi, . . . , ht, and gl(nl, . . . , n t ) = o for any other g-tuple of integers. Then

I_[ gl(nl, �9 nt), gl(B) = (,1 ...... t) e zt " "

and this is a Lie algebra gradation of gl(B). The elements h l , . . . , h e , e l , . . . , et, f l , . . . , J r are linearly independent in gl (see [IO], [I5] ). In particular, dim I?a(B)=t. The space gl(o, . . . , o, I, o, . . . , o) (resp. gl(o, . . . , o, - - i , o, . . . , o)) is nonzero and is spanned by e~ ( resp . f ) ; here :L I is in the i-th position. Also, each space gl(nl, . . . , nt) is finite-dimensional. There is clearly a Lie algebra involution ~ of gl interchanging e i and f~ and taking hi to - -h i for all i = I , . . . , g . The involution ~} takes each space gl(nl, . . . , nt) onto g l ( - -n l , . . . , - - n t ) .

The Zt-graded Lie algebra gl contains a unique graded ideal rl, maximal among those graded ideals not intersecting the span of hi, ei, and f~ ( I < i < g ) (see [IO], [15] ). We let g(B) be the Zt-graded Lie algebra gl(B)/rl . The images in

g(B) of hi, % f i , gl(nl, . . ., he) and Ih(B ) will be denoted by hi, % J~, g(nl, . . . , nt) and t)(B), respectively.

We let Di(i < i < g ) be the i-th degree derivation of g(B); that is, the derivation which acts on g(n:L, . . . , nt) as scalar multiplication by n i. Then D 1 , . . . , D t span an g-dimensional subspace bo of commuting derivations of g(B). Let b be a subspace of bo. Since b may be regarded as an abelian Lie algebra acting on the b-module g(B) by derivations, we may form the semidirect product g " ( B ) = b • (e for "extended") with respect to this action. We note that I?*(B)=b| is then an abelian Lie subalgebra of g*(B), and Ig*(B) acts on each space gl(nl, . . . , nt) via scalar multiplication. We define a l , . . . , atei)*(B) *, the dual space of tl*(B), by the conditions [h, 4]=a~(h)e~:. for all hell(B), and all i = i , . . . , g . We note that aj(h~)=Bij, for i , j = I , . . . , g (see (3.1)).

We now make the basic assumption that ~1 is chosen so that a l , . . . , a t are linearly independent. This is always possible, as we may take b =b0. (In this case, we have

a~(D~)=Si~ for all i , j = I , . . . , g). However, we may wish to choose b smaller than

b0; e.g., when B is nonsingular, then b = o is a natural choice.

202


For aet)*(B)*, define

g~={xeg(B)l[h, x]=a(h)x, for all hEt)e(B)}.

Note that [g% gb]Cg~+~, for a, beD'(B)*. Also, it is clear that e i (resp. f/) spans g,i (resp. g-,i) for each i = I , . . . , t and that for all ( n l , . . . , n t ) e Z t,

g(nl, . . . , nt) C ~ lal-~" "'-t'~t~t*

Indeed, since a l , . . . , a t are linearly independent, the inclusion is an equality, and the decomposition

g(B) = (-1,..~tlzzt g(nx, . . . , nt)

coincides with the decomposition

g(B) = =e~B,g=" b()

We define the roots of g(B) (with respect to ~e(B)) to be the nonzero elements ae[ge(B) * such that g~#o. We let A(B) denote the set of roots, A+(B) (the set of positive roots) the set of roots which are nonnegative integral linear combinations of al, . . . , at, and A (B) = -- A+ (B) (the set of negative roots). Then

A(B) = A+ (B) u A (B)

(disjoint union), g~

g(B)=I)(B)| H ga| I_[ g% a~ A+(B) affAL_(B)

and dim g - ~ = d i m g% for all acA(B). We call the elements a l , . . . , a t simple roots (this being relative to our choice of A+(B)).

We let RCt)~(B) * be the subspace spanned by A(B) (so a l , . . . , a t is a basis for R). Then the restriction map R-+I)(B)* is an isomorphism if and only if B is

nonsingular. Now, since B is symmetrizable, there are positive rational numbers q l , . . . , qt

such that diag ( q l , - - - , qt) B is a symmetric matrix. We then define a symmetric bilinear form a on R by the conditions ~(a~,aj)=q~Bij , i , j = i , . . . , t . Note that q~=~(a~,ai)/2 for each i. Set h'~i=qihi.=cr(ai, ai)h~/2 in D(B), for i = I , . . . , g .

t t

Then for a~R, with a = ~ ~iai, [xiek, define h~= Y~ ~xih'a~ in I)(B). Transfer ~ to i=l {=I

a symmetric bilinear form (again denoted by a) on D(B), determined by the conditions z(h'~, h'~j) = z(a~, a~), for all i , j = ~, . . . , t. Then a(h~, h;) = ~r(a, b) for all a, b z R . Also, cs(ai, aj)=aj(h'~i), for all i , j = I, . . . , t, so that

r h'b) = ~r(a, b) = a(h'b) = b(h;),

for all a, bsR. The form a extends to a symmetric, g(B)-invariant bilinear form

(again denoted by z) on g(B), such that [x , y ]=~(x ,y )h '~ , for azA(B), xeg% y e g - "

(see [io], [i5] ). In particular a ( e ~ , f ) = 2 / e ( a i , ai) , for i = I, . . . , t . Also, for

203


aeA(B), beA(B)u{o}, one has a(ga, fib)__O, unless a = - - b , and then ~ induces a nonsingular pairing between fla and g-~ (see [IO], [15] ).

It is clearly possible to extend the symmetric form . on R to a symmetric form cr on I~'(B)* satisfying the following condition: For all a e R and Xet)'(B)*, o(X,a)=X(h'~). Fix such a form cr on I~'(B)*. For each i = I , . . . , g , define the linear transformation r~ : [~*(B)*--->~e(B) *, by

( 3 . 3 ) r,(x) =

We let W = W ( B ) (the Weyl group) be the group of linear automorphisms of ~*(B)* generated by the reflections r~, i = i, . . . , l. The following is proved in [8], Prop- osition (2. IO) (and is due to Kac and Moody).

Proposition (3.3)- - - The form (~ on D*(B)* is W-invariant.

Examples: (i) We let A be the l • Cartan matrix associated in the usual way, with a k-split, simple Lie algebra g of rank g. (We will call such an A a classical Cartan mat r ix- -we note that a classical Cartan matrix is always symmetrizable: see below.) Then it is a theorem of Serre (see [I9] , Chapitre VI, p. I9) that

g - gl(A) = g(A).

For a classical Cartan matrix A, when there is no danger of confusion, we write A, A+, I), and 9 for A(A), A+(A), I?(A), and g(A), respectively. This notation is consistent with that in w 2. Since A is nonsingular we may (and do) take b----o, so we have tj~(A)=I~, ge(A)=g. Also, we write E l , . . . , Et, F 1 , . . . , Ft, I - I 1 , . . . , H t , for the generators el, . . . , e t , f l , . . . , f t , hi, .... , h t, respectively. The Kac-Moody construction of a Lie algebra corresponding to a symmetrizable Cartan matrix B, automatically gives a choice of simple roots. In the case of a classical Cartan matrix A, we denote these simple roots by ~1, - . . , at, and we denote roots in A by Greek letters e, [~, 7, . . . . I f ( , ) denotes the Killing form (on g, I?, and I?*), and if qi = (~, ai)/2, then the matrix

diag(ql, . . . , qt)A

is symmetric. Letting ~ denote the corresponding inner product on R ( = I f , in this case) we see that , and ( , ) agree as forms Oil I?*, I?, and g. We let e0 denote the highest root of g (with respect to our choice el, . - . , ~t of simple roots), and we set a t + l = - - ~ 0 . Throughout this paper, we will use A to denote a classical Cartan matrix.

(ii) We let A ~ ' = (A~j)~d = 1 ..... t + 1 denote the symmetrizable Cartan matrix, defined by the conditions

2( i, g,

We call A the affne Cartan matrix (associated with A) and g(A) an affne IAe algebra. The matrix A has rank t, and if we take b to be the k-span of D = D t + I , the

204

T H E A R I T H M E T I C T H E O R Y O F L O O P G R O U P S ~9

+ 1-st degree derivation, then our basic assumption, that the simple roots a,, . . . , at+ * are linearly independent on I)'(A), is satisfied. We therefore make this choice for b, and define fl'(A), t)'(A) accordingly.

Our next goal is to show, using a result of Kac and Moody, that g(A) is exactly our central extension ~, of the loop algebra ~'. Recall that veZ~('~, k) is the cocycle defined at the beginning of w 2, and

o-+k-+g~-+ fl-+o

is the corresponding central extension, constructed as in w I. In particular, ~,----~@k, and we let q~ : ~ - + ~ be the injective linear map defined by ~0(~)=(~, o), ~ . Then q0 is a section in the sense that

(3.4) ~ o qo = identity.

We let i~ • C ~ denote the subalgebra

iY• H g~'| I I t"| ~ A + ( A ) •

where Z+ denotes the nonzero, positive integers. Since v restricted to iY • • • is zero, the map q~ restricted to if• is a Lie algebra monomorphism.

We fix elements E+~fl -~=oCg, so that if H=0=[E=., E_=~ then

~(H,.)----2(~, %)(%, ,0) -1,

for all ~ A (our choice of E• is consistent with our choice of E, in w 2; these choices will be further normalized in w 4). We set

( 3 . 5 ) ~,=q~(E,), 3~=q~(F,), h, ---- q~ (H,) , i = i , . . . , t,

and we set

(3 .6) g t + * - q~(t| ~0), J~+* =q~174 t/t+, = (-- H~~ ~(%, %)-1).

We now state a theorem of Kac and Moody (see [io], [16]):

Theorem (3 .7) . - - There is a surjective Lie algebra homomorphism ~ : g(A) --+~, and is uniquely determined by the conditions: ~(ei)=Ei , ~ ( f~)=Fi , 7:(h~)=Hi, for i = I , . . . , l

and ~ (e t+ l )= t | , ~ ( f t + l ) = t - l | 1 7 6 and ,:(ht+l)=--I-I~0. Moreover, the kernel c t

of rc is a one dimensional subspace of I)(.~). Indeed, i f % = 2~ mio~i (the mi being g ~=1

nonnegative integers), and i f we set h / = ( Y~ m~h'~)q-h',e§ ), then hi spans c. Finally, r is the center of g(A). ~=1

Of course the elements e~, f i , h~ in g(A), denote the generators given by the Kac-Moody construction described earlier.

Now, in particular, Theorem (8.7) implies that we have a central extension

(3.8) �9 ~ 7 g

o-+ c-+ fl(A) ~ ~'-+ o.

205

3 ~ H O W A R D G A R L A N D

Hence, by the universal property of the central extension (2.37), we have a unique morphism from the central extension (2.37) to the central extension (3.8); i.e., we have a commutat ive diagram

, k , f i , , g , o

(2) " > c > g > ~' > o,

( 3 . 9 )

where I denotes the identity, + is a Lie algebra homomorphism, and of course the map k-+r is just the restriction of + to k.

Lemma (3. io ) . - - We have

+(~) = e,, +(j~) =f~,

for i = I , . . . , t-Jf-I.

A and + ( h~) = h~ ,

Proof. - - By the commutat ivi ty of the diagram (3-9), we have

( 3 " I I ) 7~(+(ei) ) = ~ ( ~ )

r:(+(i,)) =:U(h,), i = I , . . . , g + I .

On the other hand, from the definition (3.5) and (3.6) of ei, ~ , hi, the fact that ~oq~=ident i ty , and from the conditions r~(e~)=E~, r ~ ( f ) = F i , r~(/~.)=I-Ii. , i = I , . . . , t , and =(et + l ) = t| E_=. , n ( f t + l ) = t - l | =(ht+~)=--H~~ , of Theorem (3.5), we see that

(3. x2) ~(e , ) = "~ (~)

7:(/~.) = U(h,), i = x , . . . , l + x .

Hence, from the exactness of (3.8), we have, comparing (3.11) with (3.I2):

(3. I3) e~ = +(~) mod c A---- +(~) m o d e h, = +(hi) mod r

But then a direct computa t ion shows: I

Ei,, l, i = i , . . . , t + i .

(the crucial computa t ion is to show that h t + l = [~'t+1,9~+~]). Hence, since r is contained in the center of g(,~), we have:

I ^ I ~(?i) = ~ [+(hi), +(~)] = ~ N , el] = ei,

and similarly +(3~)=f~, for i = i , . . . , g + I . But then +(h~)=d~([~ , f~] )=[e~ , f ]=h~, i = i, . . . , g + i, and this completes the proof of L e m m a (3. IO). �9

206


We can now prove:

Theorem (3. I4). - - The map + is a Lie algebra zsomorphism. Thus the central extension (3.8) is the universal covering of "ft.

Proof. - - Since, by construction, the algebra ~'(A) is generated by the elements ei, f i , and hi, i = I , . . . , t - t - I , Lemma (3.Io) immediately implies that + is surjective.

On the other hand, by the commutativity of (3.9), kernel + C kernel ~ = k . I f ke rne l++o , then ~((o, i ) ) = o , and hence (see (3.6)) we have

+(ht+l) = ~b((-- H~., 2(%, %)-~)) = ~b((-- Ha., o)).

But --Ha0 is a linear combination of H1, . . . , Ht, and hence ~b((--H~o, o)) is a linear combination of the elements O/((H1, o)), . . . , ~((Ht, o)); i.e., h t + l = ~ ( h t + l ) is a linear combination of the elements h i = d/(hl), . . . , ht = +(ht). But since the ideal rl C gl(A) does not intersect the span of the ei's, f / s , and h/s in gl(A), and since these elements are linearly independent in gl(A) (as we noted earlier--see [io], [i5]), we have obtained a contradiction. Thus + is an isomorphism. �9

4. The Chevalley basis in the universal covering.

In [7], w 4, we introduced a Chevalley basis in the algebra fi(A). By means of the isomorphism d? : ~ - + g ( A ) , we may then pull this basis back to a basis in ~ . In fact it is quite easy to describe the "pull back" basis in ~ , and then to compute the bracket relations using the cocycle v. We wish to give the description in the present section.

First, we begin with a simple Lie algebra g over the field 13 of complex numbers. Indeed, let g have rank t, and let A be the e• (classical) 13artan matrix corresponding to g. Then g = g ( A ) is the Kac-lVIoody algebra constructed from A, as described in w 3. From the Kac-Moody construction (in this case, the Serre construction), we have a Cartan subalgebra D =I)(A), the set of roots A (relative to I)) and a given set of simple roots eel, . . . , a t. We then know that g(A) has a Chevalley basis

elements E~ega, (see Steinberg, [2I]). Thus, for each 0teA, we have nonzero H~eI), and these elements have the bracket relations:

[Ea ,E ~ ]=H~, aeA

[E~, E ~ ] = ~ : ( r + I)Ea+~, if ~, ~ and 0~+~eA, and

0c-- r~, . . . , ~, . . . , 0~ + q~ is the }-string through a,

[E~, E~] = o, otherwise,

[H~, E~]-- 2(~, ~) E~, 0~, ~eA.

207


For the Chevalley basis of g, we take the set:

(4.2) 'V={E~L~u{H1, . . . , Ht}

(where ~ = H a p i = i, . . . , t ) . We note that, in particular, this basis has integral structure constants.

Now, since the isomorphism + sends b'~ to ei, and j~ to f , it is easy to determine the subspaces of ~, which correspond to the root spaces of g(A). Moreover, one can easily compute the derivation of ~ which corresponds to the l + 1-st degree deri.

e~a

vation D of g(A). t

I f al, . . . , at+l~i)*(A) * are the simple roots and if % = ~ m~0~ i is the highest ' ~ = 1

root of g (relative to the choice 0~1, . . . , 0 9 of simple roots) then we let t~D*(A) * be the element

l

~=( ~ mr +at+ 1.

t

Also, if 0c= 2~ n~0q~A, we let a(~)EIg*(A) * be the element i = 1

g

a(~) = Y~ n~a i.

We then have (see [7], w 2):

(4.3) A+ (A) = { a ( ~ ) } ~ A~IA)U{a(~) + n~}~,e,x(A),,~ez+W{m}nEz+,

where Z+ denotes the set of all (strictly) positive integers. We let

• :{a(~) + n~}~ ~ A(~),.~.., t ~

AI(A) ={m}nez,n*0,

and call the elements of Aw(A ) Weyl (or real) roots, and the elements of AI(A ) imaginary roots. We let Aw,+(A ) (resp. AI,• denote the set of • Weyl (resp. • imaginary) roots.

Then, if we identify g(A) with ~, by means of +, we have

(4.4) g~{~)+"~=(t"| ~, o), aEA(A), n~Z,

g"~=(t"| o), n~Z, n4=o.

Also, we have

~(h) = { o,i, h=I~,h~I~(Y')

and thus /

n = { (4.5)

Indeed, (4.3), (4.4), the commutativity of

scalar mult. by n, on fl~(~}+"t

scalar mult. by n, on g"~.

(4-5) follow easily from Theorem (3.7), Theorem (3.I4), and the diagram (3.9) (also, see [7]).

208


Remark. - - One can deduce the existence of D on ~ directly from the universality of the central extension o-+k- - -~f i , -+~o , vation D O : ~ ' ~ , defined by Do(t" |174 lift of D O on ~.

We define elements ~ in g~, aeAw(A), by

~ =( t" | o), a = a ( ~ ) + n ~ .

For

and from the existence of the deri- neZ, xeg. Thus D on ~ is the

neZ, n+o, we then define elements ~(n), i = i, . . . , to, in g"~ by

~,(n)=t" | neZ, n#o , i = I , . . . , to.

Finally, we let

(so hi=i,, (4.6)

h i = (Hi. , o), i = I , . . . , t o

2 ( % , % ) - ' )

i = I , . . . , t o + I ) . Then the set

. . . . . . . . . . t

is a basis of ~ , = g(.~), and is called a Chevalley basis. One can of course compute the structure constants directly, using the cocycle v.

In any case these structure constants are computed in [7], w 4, and are seen to be integral. Indeed, the integrality of the structure constants boils down to showing

that - - is an integral linear combination of hi, . . . , ht+l , when ei is a long root.

This is part of the proof of Lemma (4. i o) of [7], and is derived by examining root diagrams.

Remark. - - The element (o, i) in ~ , = ~ | corresponds to h~=g(A). However, if we replace the Killing form ( , ) by the form ( , ) = v ( , ), where v > o is chosen so that (H~, I - t~ )=2 for ~ a long root, if we define the cocycle v' by

":'(u| v| , v ) ( x , y ) , u, w k [ t , t - l] , x , y~g ,

(compare with w 2, after (2.2)), and if we consider ~ , , = g | in place of ~ , then (% i)e~, , corresponds to 2h~/(e, 00, e a long root. Thus it is natural to construct our Chevalley basis in g(A), by using ~,, and ( , ) in place of ~, and ( , ), respectively. Roughly speaking, a Chevalley basis in ~,, is constructed by lifting an obvious choice of Chevalley basis in ~, and then adjoining the element h / + l = ( - - H ~ . , I).

Notational Remark (4.7)- - - From now on we use g(A), and not ~,, to denote the universal covering of ~'. Indeed, identifying 9(,~) with ~ by means of +, and using the commutativity of (3-9), we see that r: : g(.~)--+~ corresponds to ~ : ~,-+~. We thus write ~ in place of 7:; i.e., in sum, we let

,,, ~o (4.8) o-~k~ g(A) - + ~ o

now denote the universal (and Kac-Moody) central extension of ~.

209

5


We now define subalgebras u• by

~ a G A+(A)

On the other hand, in w 3 (just after (3-4)), we defined subalgebras ~'• Since kernel gC t)(A), by Theorem (3.7) (we have set g = = ) , we know that the restriction gm of g to u• is injective. Moreover, from our description of the root spaces in (4.4), we see that Image g =L =if+; i.e., g• defines an isomorphism

: u * ( X )

We will write u(A) (resp. g) for u+(A) (resp. R+), whenever convenient, and we will identify u• with ~• by means of co , whenever convenient. Thus, we may, for example, regard tn| ", n>o, ,eA(A), as the root subspace g~, a-=a(,)+m, of u(A), and we may regard ~ as an element of g + (namely, ~=tn |

We let gz(A) (resp. gz(A)) denote the Z-span of the Chevalley basis �9 (resp. of the Chevalley basis Re). Then gz(A) (resp. gz(A)) is an integral subalgebra of g(A) (resp. g(A)) by [7], Theorem ( 4 . I 2 ) (resp. by [OI], Theorem I, p. 6). If R is a commutative ring with unit, we set

ga(A) = R| z gz(A),

gR(A) = R| gz(A).

We let R[t, t -~] denote the ring of Laurent polynomials ~oS<iq~tr (finite sum, with

i0 and i~ allowed to take negative values) with coefficients q~ in R. We let

g"~R = R[t, t -~] | gz(A),

and observe that

We note that ~ induces by restriction, a surjective Z-Lie algebra homomorphism

~,. : gz(A)-- ,g~.

By (4.5) and the remark following, we then see that

D(gz(A)) C gz(A),

Do(gz) C g'z.

Hence, for any commutative ring R with unit, the operator D (resp. Do) on gz(A) (resp., on ~'z) induces an operator on fll~(-~) (resp., on g'R) which we also denote by D (resp. Do).

We let u~(A)=gz(A)t~u• and iI~=~'zrMI • We then have e ~ l t ~ a

coz(Uz~ (A)) =iY~.

210


We let ~z(A) denote the Z-span of hi, . . . , he+ 1 and Dz(A) the Z-span of H1, . . . , H t. We have

g,.(X) = u~ + (X) e ~, (7,) �9 u~ (Y~),

Moreover, the algebra u~(A) (resp. iY~; resp. Dz(A); resp. Ikz(A)) is a Z-form of u• (resp. ~'• resp. t)(A); resp. D(A)). We let

u ~ ( X ) = R |

~• = R| ~ R

~)a(A) = R| z ~)z(A).

We then have direct sum decompositions

g~(X) = u + (X) | O,~(X) eu~(X), ga~ = i f + | ~)R(A) |

Moreover, COz:gz(A)-+g z induces a surjective Lie algebra homomorphism

(recall that gR(~i)=R| g ~ = R | where

~a (OR(X)) ----I)a(A) ,~ •

kernel G C ~(X) . ~ ~ •

Thus co R restricted to u~(A) is an isomorphism which we denote by ~0 R. We identify u~(A) with if• by means of ~ .

Finally, when there is no danger of confusion, we will write u~ for u~(A), and (more simply) UR for u+(A), and fir for U + .

5. Completions.

For ieZ,

Of course,

and

we define

g~(X), ={x~ g~(X) l D(x) = i~},

'~a,,= { xegRl Do(x) = ix}.

gR,~ = t~| gR(A), i eZ ,

gR(X), = t' | iEZ, i # o

211


(recall we have identified u~ (A) with it +, by means of co R ). the direct sum decompositions

(5 "I ) fR(~) = i I~z fR (A)i ,

and C0R(fR(A),)=gR, ~. I f XefR(A ) (resp. XefR), if we write

O f course, also, we have

x = ~ z xi (finite

and sum), with xi+fR(A)i (resp. xie~R,i), if we set io=inf{jlxj,o}, (5.2) Ixl =2-'.,

(if x = o, we set I xl = o), then we call [x] the norm of x. We remark that the choice of 2 in (5.2) is a rb i t ra ry- -any p > I would do. We have

(5.3) Ix+yl<sup(lx[, [y]) and I[x,y]l<~lxllyl for x,y+fR(X ) (resp. gR).

I f R has no zero divisors, we have

(5.4) Ir.xl = / Ixl, rcP~--{o}

[o, r = o .

We note that

(5.5) l~R(x)l<l~1, x+g~(7~), so that ~R is norm decreasing, and hence uniformly continuous. Moreover,

(5.6) I~,~(x)l=lxl, .~u~. We let g~(X) (resp. ~ ) denote the completion of fiR(A) (resp. gR) with respect

to the norm [ 1. Then g~(A) and g"~ have induced Lie algebra structures and the norms 11 extend to these completions. Moreover, the extended norms also satisfy (5.3) and (5-4) (the latter, provided R has no divisors of zero). Also, by (5.5), ~R has an extension to a Lie algebra homomorphism

(5.7) ~% : g~(X) - + ~ ,

and (5.5) is still valid for this extended homomorphism (so the extended ~1~ is continuous). We let u~ (resp. ff~) denote the closure of 11R (resp. iYR) in f~(A) (resp. ~c). We then have the direct sum decompositions

f~(X) = u~+ N(X) + u~,

and the extended homomorphism g~ (in (5.7)), when restricted to u~, defines an isomorphism

212


This last assertion follows from (5.6). I t then follows that ~ in (5.7) is surjective and

kernel(e~ : ' ~ g~(A) -+ glt) ~~

is contained in I?tl(A), and is equal to kernel(o~R: ga(A)-+g~).

Remarks (5.8). - - (i) We let ~q~R=R[[t, t -1] denote the t-adic completion of R[t, t - l ] ; i.e., 5r R is the ring of all formal Laurent series

(5"9) ~ = ~ q~ ti, q~eR, i > io

where the sum on the right is allowed to be infinite. We note that ~ is isomorphic to g .~ (A)=s174174 ). Moreover, if for a as in (5.9), we have q~04:o, we define l a l = 2 -~' (we set I o1=o) . Then ~C~'a is the completion of R[t, t -1] with respect to this norm, and this norm on ~R induces our norm I I on ~ . (ii) From the proof of L e m m a (4.Io) in [7], we have

[ 2hi =ht+~+Zk&, k, EZ.

(~0, ~0) Therefore, the set

{hi, . - . , ht, 2h:/(~o, ~0)}

is a Z-basis for I)z(A ). I t then follows that I| %)) in I)a(A)=R| spans kernel(co R : g~(A) ~~ ~ ~ ~ -+glt)=kernel(o0R: g~(A)-+glt). I f we call this kernel CR, we have the two central extensions

(5" IO) a) O-+C~-+gR(A)---~gR-+O,

0 0 b) o-+CR-+gR(A) -+gR-+o.

Remarks ( 5 . I I ) , - - (i) I t follows from Remarks (5.8) (ii), that (identifying g(A) with fi ,=~' | over a field k of characteristic zero) we have gz(~)=~ 'z | where (o, i) in ~'z| corresponds to the element 2h~](%, %) in gz(A). Thus BR(A) =~'R| g ~ ( A ) = ~ | and we have naturally defined sections ? :g'R-+ BR(A), q 0 : g n ~ g ~ ( A ) , each defined by q0(x)=(x,o) (xe~'R, fli~, respectively). The first section is just the restriction of the second, and hence the cocycle corresponding to q0 on gR is just the restriction of the cocycle corresponding to q~ on gR. The latter is the cocycle -% defined by

(~0, ~0) "%(u| v | u, Ves x, yegz (A ). 2

(ii) I f R = k is a field of characteristic zero, then the central extension (5. Io) b) is universal in the category of central extensions

(5. x2) r 7 ~ l P ~ e

0 - - + / ~ - - + fl --> fiR---> 0 ,

213


which split over a sufficiently deep congruence subalgebra of ~ . More precisely, we set 0C ~ equal to the ring of formal power series

~ = Y~ qit i, qi~k, i>o

and consider gr C g.~,(A)= ~'~.

By the congruence subalgebra o f ~ of level n, we will mean the subalgebra of all xeg, (A) , such that x---o mod t ". We then consider the category of central extensions (5.12) which split over the congruence subalgebra of level n, for some n, and we claim the central extension (5. lO) b) (for R = k ) is universal in this category. The argument is essentially the same as in w 2. There is just one additional point. Thus, define {u, v}, u, VEs exactly as in (2.5). Then in order to prove (2. I8), one can argue exactly as in w 2, once one proves that { , } is continuous on ~ • s162 in the sense that if ui-+u, vi-+v, are convergent sequences in ~ (relative to the t-adic topology) then {ui, @ = { u , v}, for i sufficiently large. To see this, one first notes that there exists an integer n > o so that {u ,v}=o , whenever u, ved) and u , v = o m o d t ". But then the desired continuity is an easy consequence of this and the second identity of (2. I2), for u, v, w e ~ .

6. Representation theory.

In [I I] , Kac introduced highest weight modules for a Kac-Moody Lie algebra g(B) associated with a generalized t • Cartan matrix B. Kac's modules are described in [8] and [7], w I o. We refer to the latter as a general reference. Briefly, the construction is as follows: We let

u * ( B ) = I_I g~ aEA•

p' = I)e(B) | (B),

so u• and pe are subalgebras of g~(B). For any Lie algebra a over a field k, we let ~//(a) denote the universal enveloping algebra of a. We say XeI)'(B) * is dominant integral, in case, (i) x(~)eZ, i=~, . . . ,e , and (ii) X(hi)>o , i = i , . . . , t . I f X only satisfies the first condition, we say X is integral. We let D = D ( B ) denote the family of dominant integral linear functionals in I)"(B) *. For XeD, we let M(X) denote the one-dimensional p%module, with p%action defined by

We let

Left ture.

h.v=X(h)v, hEIk"(B),

~ . v = o , ~eu+(B),

~ " = ~ ( p ~ ) , and set

V M(x/--- ~(g~ (B))| M(X).

multiplication gives V ~(x/ a ~

v M(X),

w M(X).

(or equivalently, g"(B)) module struc- As a g~(B)-module, V ~(x/ contains a largest submodule not intersectiong the

214


subspace I| CV u(x) nontrivially. We let V x denote the quotient module of V M(x) by this submodule. We call V x the highest weight module with (dominant

integral) highest weight X.

We now focus on the case when B = A, the affine Cartan matrix associated with the classical Cartan matrix A. From now on, we take g(A) to be the corresponding Lie algebra over C. We also fix a highest weight vector v o ~ o (by definition, a highest weight vector is non zero) in the highest weight module V x. Thus we fix v+o in M(X), and let v 0 denote the image of IQV in V x.

In [7], we introduced a Z-form @z(A) of 0~(g(X)), and the Z-form

= % ( X ) , v0

of V x. For pteI)e(.~) *, we let V~ x denote the subspace

[ h =

I f VX~+o, we call ~t a weight o f V ~ and we call V~ the weight space (for the weight ~). We call nonzero elements of V~ z weight vectors (of weight ~). Then (see [8]), V x is

a direct sum

V x = II V x ~(~)*

of its weight spaces. We know (see [7], Theorem (11.3) and its proof) that Vz x is the Z-span of an admissible basis, i.e., of a basis ~ of V x which is the union of its intersections f ~ = f~c~V~ with the weight spaces o f V x (see [7], w i i, Definition (I I . 2)).

X X We note that if we set V ~ . z = V j ~ V ~ , then it is a consequence of our last assertion that

(6.x) VzX= H V x ~t ,Z �9 ~ l ie(A)*

For a commutative ring R with unit, we set V ~ = R | Vx and

(6. ~) V x~,R ---- R| V~X,z .

From (6. I) we then have

N )t (6.3) V~= II V~, R. tt ~be(A) *

Of course, in this direct sum, it suffices to let Ez vary over the weights of V x.

Similarly, the Z-module t)z(.~)=dfgz(~)nI)(,~) spans I)(.~), and for root a~A(A) the Z-module g~----~fgz(.~)c~g ~ spans g~. Moreover, we have

X I)z( )

a A(X),

and gz (A) = Dz(-~) | e~(~) fl~'"

each

215

4 ~ H O W A R D G A R L A N D

I f R is a commutat ive ring with unit, and if we set

DR(A) = R| I)z(a) and g~ = R| z g~,,

then gl~ (A) = I)a(~*) | e[~A(~)g~"

We also note that if a is a Weyl root then g~, as an R-module, is isomorphic to R, while if a is an imaginary root, then g~ is a free R-module of rank t. When considered as an R-module, t)R(A ) is free, of rank t q - I . We also note that

g~(~)~ = II g ~ | g~, if j # o, a = a(%) + # ' =

G A(A)

(6.4)

while

(6.5) G A(A)

Alternatively, we have

g~(A)0 = ~ ~(g~(A)).

We note that gR(~.) acts on VR x.

Proposition ( 6 . 6 ) . - - - L e t {x,}n=a,z, a ..... be a Cauchy sequence in gR(A) (relative to the

norm I [ of (5.2)) . Then for each WVR x, the sequence of elements x , . v in V~ is eventually

constant.

Proof. - - As we noted in [8], w IO (and as is easily seen from the definitions), each weight bt of V x is of the form

g + l

(6 .7 ) bt=X-- ~ k~ai, k~>o, kieZ. ~ = 1

Also, we set g~(A) = ~z(A)|

=O,.(E) e z D D~(A) = R|

and note that if X(D)eZ, then VR x is a module for the R-Lie algebra g~(A). From

now on, we assume X(D)eZ (so D acts on Vz~). We set

V~ = { veVXl D(v )= (x(D) + j ) v } ,

V x~,~--R| j>o,_ j e Z .

We say that the weight ~ of V z has D-level j , in case then, thanks to (6.7) ,

(6.8) VX= II = weight '

of D-level j

t + l

~ = X - - ~ kiai, and - - k t + l = j ; i = 1

216

and therefore

(6.9)


IIV , j<o

V~= C| uv z,

j_<0 '

Indeed, the first equality of (6.9) follows from (6.8) and the fact that V x is a direct

sum of its weight spaces. The second equality follows from (6.8), (6. i), and the fact x C| The third equality follows from (6.1) and (6.8), and the that V~ = . .

fourth from the third. Also, we have

(6 .xo ) gR(A),. VxR C V~+j,R,

as one can check directly.

To prove the proposition, we may, thanks to the fourth equality of (6.9) ,

assume raVeR for some j < o (in Z). Then, since {X,}n=I,2, 3 .... is a Cauchy sequence, we may choose n o so that for all n ~ n o we have:

But from the last equality of (6.9) , and from (6. IO), we have

flR(A)~.V~R = o if i + j > o .

Thus, for n>no,

Xn. V : Xno. V,

and this proves the proposition. �9

I f x~g~(A), let {xn}n=l,z, 3 .... , be a Cauchy sequence in ga(A) such that lirno~ xn=x . Then for wV~ the proposition implies that x,. v is eventually constant.

It is easy to see that this constant value is independent of our choice of Cauchy

sequence converging to x. We set x. v equal to this constant value, and in this way we obtain a ~ (A) -modu le structure on V~. We let

r:~ : g~(X)-+End VR x

denote the corresponding representation of g~(A).

Notational Remark. - - We also denote the restriction of =~ to gR(A) by =x.

When R = G , we write ~x for r:xa and g*(A) for g~(A).

Proposition ( 6 . i i ) . - / f XeD and X(~)4:o for some i = i , . . . , t @ i , then

~x : g~(~) ~ E n d V x is injective.

217 6

4~ H O W A R D G A R L A N D

Proof. - - Choose a basis of V x consisting of weight vectors and order this basis so that the vectors in any given weight space appear in succession. I f xegC(A), we may write x as

x = x ~ + Z _ x a, a G A(A)

where x0eD(:), and xaeg a, aeA(X), and all but finitely many x a with aeA_(A) are non-zero (however, infinitely many x ~ with aeA+(A) may be non-zero). Relative to the basis of V x which we have just constructed, 7:X(x) is represented by an infinite matrix with a natural block decomposition, each block corresponding to a pair of weights. Then the transformations r:X(x~), r:X(x~), aeA(A), correspond to various mutually distinct blocks of the matrix representing ~X(x). Thus, if ~X(x)=o, then r:X(x~)=o and r:X(x")=o, for all a~A(A). Thus, if ~z is not injective, we may assume r:X(x)=o where either x~t?(A) or xeg ~ for some a~A(~), and x+o. We now show:

(6.I2) I f r~ x is not injective, then 7:X(x)=o for some non-zero element x

in ga, aeAw(~).

By the above remark, we may assume r~X(y)=o for some y4=o and either yeI?(A) or y e g a, aeAi(A ). Also, hi is a nonnegative linear combination of hi, . . . , ht+ 1. Hence h~6kernel ~x, since X(~)>o for all i = i, . . . , g + I, and X(~)>o for some i = I, . . . , g + i. I t follows that if either yeD(A ) or y e g ~, aeAi(A), y4~o, and =X(y)=o, then one of the elements [y, e~], i = I, . . . , g-t- i, is non-zero. Since (kernel nZ)ng(A) is an ideal, we obtain (6. I2).

Thus, assume x is a non-zero element in g~, a~Aw(A), and r~X(x)=o. We will show r~ x is then identically zero, and this contradiction will prove the proposition. First, we will show we may take a to be a simple root a=a~, for some i = i, . . . , g + I. Toward this end, we first recall that in [7], w 6, we defined for each beAw(A), an automorphism ~ of gz(A), by

= exp (ad ~b) exp (-- ad ~b) exp (ad ~b).

Of course we may also regard r"~ as an automorphism of g(A), and then by

Lemma (6.9) in [7], we have

(6. b ZXw(7 ), where hb=2h'b/a(b,b), for beAw(A ). On the other hand, for each i = I , . . . , g - t - r , we define a linear automorphism

by

(6. I4) X(r,(h))----(r,X) (h), hG If (,~), Xe I)'(A)*.

A direct computation, using (3.3) (i.e., the definition of r i on t)*(A) *) then shows that

r,(h)=h--a,(h)h, i = : , . . . , g-t- r, heD'(.~ ).

218


Comparing with (6.I3) , and setting ~---=~i, i = I , . . . , g + I , we see that

~(h)=rdh), i = i , . . . , gq-i , h~b'(,~).

Thus, if w = r q . . . q k e W , and if we set ~ = q ~ . . . ~ k , then

( 6 . 1 5 ) ~(h) = w(h), h~be(~k), From this, from the fact that ~ preserves flz(,~), and from the fact that ~b is primitive

in gz(A), for each beAw(A), we easily obtain

(6. x6) ~(~b) = • ~w(b), beAw(~')

(here, we only need ~(gb)=g~(b), but we will use the stronger assertion (6. i6) later

on). Now we are assuming 7:Z(x)----o for some non-zero xe~ a, aEAw(A ). We

choose weW such that w(a) equals the simple root a~. Then, by (6.I6), we have ~(x)eg ai. On the other hand, =Z(x)=o implies ~z(~(x) )=o , since kernel ::z is an ideal in 9(,~); i.e., we have shown that if kernel ~:x# o, then for some i = i, . . . , t + i, we have 9 ai C kernel 7: ~.

We let 0C{I, ...,[-~-I} be the subset of a l l j such that 9aJCkernelT: x. We let 0' denote the complement of 0. We shall now prove that 0 is empty. I f not, we may choose ie0, j e0 ' , such that

(6. x7) .~j # o

(we have assumed g = g ( A ) is simple). We let [0] denote the set of all roots aeA(A) which are linear combinations of the a,,, with meO. We set

J = B g~ a G A+(A)-- [01

and we note that J CkernelT~ ~. But by (6.I6) and (6. i7) we have a* 5(g ') c j .

Hence g~C kernel n~, and this is a contradiction. Hence 0 must be empty. Thus 9 ~C kernel nz, for all i = I, . . . , g + I. But then g--al =q(9~ ~) (by (6. I6))

is contained in kernel 7: z, for all i-----I, . . . , / + i . Since the elements ei, f , i = i, . . . , g + i generate g(~), all of g(~) is contained in kernel ~z; i.e., we have shown that if r~ z is not injective, then ~x is identically zero on g(A). But then ~z(/~.)=o, for i = ~ , . . . , ~ q - ~ , andhence ?~(h~)=o, i = ~ , . . . , t + I . This is a contradiction, and

hence n~ must be injective. �9

7. C h e v a l l e y g r o u p s .

From now on we make the following assumptions and notational conventions:

We let g(A) denote the Kac-Moody Lie algebra over C, corresponding to the affine Cartan matrix A. Until w 16, no restriction will be made on k, except in the first part

219

44 H O W A R D GARLAND

o f w 9- We assume ?,ED satisfies X(h~)#o for some i = I , ...,~-@I. Otherwise we continue with our earlier notation; e.g., gk(A)=k|

a For each aEAw(A ) we have defined (see w 4) ~agz , and we now also let ~, denote I| ~. Now let XED. Then we have seen ([7], Lemma (lO.4)) that for all veV x, there exists a positive integer r such that

(7.I) ~ol.v=o, i=x, . . . ,~+x. In particular, this holds for veVz z, and hence for wV~. Now, for aeAw(A), ~/n!eq/z(A ) maps Vz x to itself (see [7], w xi), and hence, for any commutative ring R with unit, ~/n! defines an endomorphism x , l) ~R(~/n of V~. Indeed, we obtain a representation r:~ of q/a(A)=a~RNzq/z(A), in V~. In particular, this is true for R = k .

We note also that a similar situation holds for the adjoint representation. As usual, for 4, ~eg(A), we set ad(~)(~)=[~, ~]. We then obtain a corresponding representation of q/(g(A)) which we still denote by ad. For example, if 4, ~eg(A), then

n-times

ad(~") (~ )= [~[~ . . . [~,, ~ ] . . . ] ] .

Then ad(~/n!) maps gz(A) to itself, and hence for any commutative ring R with unit we define an endomorphism ada(~/n! ) of gR(A). Indeed, we obtain a representation ad R of q/a(A) on gi~(A).

For s~k, we then have from (7.I) that __ n ). n I . . , )+ai(S)--ea ~>oS rck(~+,]n.), i-= I, . t + x,

is a well-defined endomorphism of Vk x. Moreover, we have

( 7 "@) ) ~ • = )~4-ai($1))~-~ai($2), 31, S~k, i = I , . . . , t - ~ - I .

Thus each Z• has an inverse (namely Z• and hence Z.,i(s) is a k-vector space automorphism of Vk x.

We let Endk(Vk x) (resp. Aut~(VkX)) denote the k-vector space endomorphisms (resp. automorphisms) of Vk x. For zeAutk(Vk x) and BaEndk(Vk z) we let A d ( x ) ( B ) = z B ) -~. We shall show that, for ~agk(A),

(7 .3) Ad(z• ~,(s)) (~(~)) = ~ ~ x ~ ">~ s 7~(ad(~+~i/n.)~) (finite sum),

sffk, i = I , . . . , t -~-I .

We note that only finitely many of the expressions ad(~:~]n!)(~) are non-zero. Thus, for beAw(A), we may define an endomorphism ~/~(s) of g~(A), by

Y/~(s)(~)= Y, s"ad(~/n!)(~), sek, i=x , . . . , g § n > 0

Analogous to (7.~) we have

(7.4) ~(Sx+S~)=q/~(s~)~/~(s~), sx, s~ek, i----x, . . . , / + x ,

220


and so ~ b ( S ) - - l = ~ / b ( - - S ) , and ~/b(s) is an automorphism of gk(A). We may rewrite

(7.3) as

(7 .3 ' ) Ad(z+, i ( s ) ) ( r :2 (~) )=r :~(~ /4_a i (S ) (~ ) ) , ~egk(A), s e k , i = I , . . . , ~ @ I .

We prove (7.3') by regarding it as a formal identity in s, and computing the deriva- tives of both sides with respect to s. The (formal) nth derivative of the left side is

(7-5) Ad(z• ~i(s)) (r:kZ((ad ~. a i ) " ( ~ ) ) ) ,

and the (formal) nth derivative of the right side is

(7.6) r:k (~• ai(S) ((ad ~• ai)" (~)))"

We see that the expressions (7.5) and (7.6) are equal when s = o . This implies (7.3'), and hence (7- 3).

Now for each i = I , . [ -~- I, we define x x . . , r~ eAutk(V~) by

r.X, = )~ai ( I ) )~_ ai ( - - I ) Zai ( I ) .

On the other hand, in w 6, we defined automorphisms ~ , beAw(~X), of gz(A). Tensoring with k we then obtain automorphisms of gk(A), which we again denote by r'~. As in w 6, we set ~----~i' i = I , . . . , t + r . Clearly

rb ~ ~b( I ) ~ _ b ( - - I )@lb( I ),

and hence (7-3') implies

(7 .7) Ad(rX) x z ,-, (~k(~))=~k(r,(~)), ~egk(X), i = ~ , . . . , g+~ .

From this and from (6. I6) we have

(7.8) Ad(r~X) (r:~(~))= -t-7~k(~ri(a)), i = I , . . . , e-~-I , a ~ A w ( i ).

I f

(7.9) we set

(7 .9 ' )

w = r~l.., r~ieW,

w z _ r.X r x - - $ i �9 �9 �9 S j "

A priori, w x depends on the expression (7.9) for w in terms of the r~'s. In any case, we have from (7.8) that

(7. xo) Ad(w x) (7:~(~a)) -= 4- x ~ rck (~wCa)), aeAw(A).

Now if aeAw(A), then a = w ( a i ) for some weW, and i = i , . . . , l + i . Thus, if we write w as in (7.9), and then consider the corresponding w x given in (7.9'), we have from (7. Io) that

(7.xx) r:~(~) = • Ad(wX) (r:~ (~,,)).

It then follows from (7. i), that we have the

221

46

Lemma (7-I2) . - - For all veV~,


aeAw(A), there exists an integer r > o , such that

Moreover, (7. I I)

for each sek.

(7.x3) ~.v=o.

Thus, if we set

( 7 - I 4 ) Za(S) Z n X n V = s ~(~o/n.) , s~k, a~ZXw(X), n~0

then the sum on the right is a well-defined element of Endk(VkX). implies

z~(s) = w ~ z~, ( • s) (w ~) - 2,

and it follows from (7.~2) that

(7. ~ ) z~(s~+s~)= z,(s,)z~(s~), a~Aw(X), s~, s ~ k ,

and of course this implies, as before, that Za(s)EAutk(VkZ), We have:

Lemma (7 . I6 ) . - - Let eeA(A). For all veV~, there exists an integerjo , so that i f

J>--Jo and a-=a(o O+f l , then ~ . v = o .

Proof. - - This lemma is a consequence of (6.9) , (6. io) , and the fact that if

w V ~ , there exists an integer Jo, so that i f J>--Jo

a = a ( ~ ) + f l , then ~eflR(A)j. �9

Corollary. - - Let geA(A). For all and a = a(,) +f l , then

(7. ~7) z~(s) .v=~,

for all sek.

In w 5, we have set 5( 'R=R[[ t , t-z]] equal to the ring of all formal Laurent series with coefficients in the commutative ring R with unit (see (5.9))- We have also, in w 5, defined a norm I I on 5r We let

r (x~XeR [ [xl<_~}

0"---- 0 - - ~ .

Then �9 is the ring of formal power series in Lf n (the "integers" of .,%~ , the set ~ is the (unique) maximal ideal of 0, and 0* is the group of units in 0.

We will now consider ~s i.e., we take R to be the field k (which is not, we recall, necessarily of characteristic zero). I f we are given ~(t)e~fk, where

( 7 , I 8 ) ~( t ) : Z qjt i, qiek, i_>i,

and if oceA(A), we define Z~(~(t))=Z~(~(t))eAutk(V~) by

(7. x9 ) X~((~ (t)) :---- )~ ((~ (t) ) = i>Hj. ;(a(~)+i,(qi)"

We drop the superscript X when there is no danger of confusion.

222


By the Corollary to Lemma (7.I6), Z~(a(t)) is a well defined automorphism of V~, and using the fact that

[~a(~)+j~, ~a(~)+j'~] = O, j , j ' ~Z ,

and using (7. I5), we obtain

(7.20) Z,(al(t))z~(a~(t))=Z,((~l(t)+(~(t)), (~i(t)e~cfk, i = I , 2, ~ A ( A ) .

The following definition is fundamental in this paper:

Definition (7 .2 i ) . - - We let G = G~ ~- G~(A) C Aut V~ denote the subgroup generated by the elements X~(~(t)), ~eA(A), a ( t ) ~ . We call G the (complete) Chevalley group over k (with respect to r:~).

For ~EA(A), and a ( t ) e ~ with a(t)4:o, we set

(7" 22) w~ z ((~(t)) = w~ ((~(t)) ~- X~(o (t)) Z - . (-- o(t)- a) Z~((~ (t)), = -1.

We drop the superscript k, when there is no danger of confusion. Also, for aeAw(A), sek, s4:o, we set

(7.2S) w~ (s) ~- w, (s) ---- 7~(s) Z- ~(-- s- ~) Za(S),

h (s) = ho(s) = wo(s)wo( ) - 1 .

Again, we omit the superscript k when there is no danger of confusion. We then have the following important

Definition (7.o4). - - We let J C G denote the subgroup generated by the elements 7~(a(t)) where either 0r ~(t)e0 or 0r a ( t )e~ , by the elements h,(a(t)), a(t)e~9*, r and by the elements hat+~(s), sek* (#*----d~k--{O)). We call J the Iwahori

subgroup of G.

Remark. - - Obviously our definition of Iwahori subgroup is analogous to, and motivated by the corresponding notion introduced by Iwahori, Bruhat and Tits (see e.g., [5]). We shall make the relationship more precise later on (see w167 ~3, ~4, below).

8. The adjoint representation.

In w 7 we defined, for each a~Aw(A), a one-parameter group of automorphisms 7~(s), s~k, of V~, and a one-parameter group of automorphisms ~ta(s),

sek, of 9k(A). In analogy with the way we defined the automorphisms )G(a(t)), ~EA(A), a ( t ) e ~ in terms of the 7~(s), we now wish to define automorphisms ~/~(a(t))

C t~ , of gk(A), in terms of the ~/~(s). We also wish to show that the Z~(a(t)) and ~t~(,(t)) are related by an appropriate analogue of (7.3').

223


First we note the following general izat ion of (7.3 ') (the proof being the same

as for (7.3 ')) :

( 8 . I ) Ad(y~(s))(7:~x(~))=r:~z(Y/a(S)(~)) , ~ g ~ ( A ) , aeAw(.~), sek .

Next, we note that relative to the no rm ] I on g~(A), defined in w 5, the automorphism ~ ( s ) , aeAw(A), sek , of g~(A), is bounded; i.e., there exists C > o so that

I ~ (~,) I < C[ ~ I, for all ~eg~(A).

Hence Y/~(s) has a unique extension to a bounded operator on g~(A), and we denote this extension also by ~/~(s).

Now let a(t)e~cf k be given as in endomorphism Y/~(a(t)) of g~(A) by

(8.2) ~(~(t)) = jH ~'.(~)+/q~).

O f course we must show that ~/~(~(t)) is well defined. i > j o , we set

~, = ( ,>~j /o I~) +/qJ) ) (~),

then the sequence :Oi, i = j o , j o + I, . . .

f rom (7.4) we have

(8 .3) ~(a l ( t ) )~ '~ ((~2(t)) = q~J:~((~l(t) -~ (~(t)),

Thus ~ ( ( ~ ( t ) ) - i = ~ ( - - ( ~ ( t ) ) , and ~ / ~ ( a ( t ) ) i s aeA(A) , a(t) e~f~.

(7 . I8) , and let e eA(A) .

But if ~eg~(A),

is convergent (relative to l I).

We define the

and if for

Moreover,

qi(t) E.~Pk, i = I, 2, eeA(A) .

gk(A), an au tomorphism of c "~

Actually, for the same reason that ~ ( a ( t ) ) is well defined, we have:

(8.4)

(8.4')

for

Let ~EA(A). I f a~(t), i = i, 2, . . . , is a sequence in 4 , and if

lim i a~(t) = a(t) ~s

(relative to the n o r m I I on ~-~k), then for all ~q~g~(A), we have

(relative to the n o r m [ [ on fl~(A)).

Similarly, f rom the Corol lary to L e m m a (7 . I6) , we have

Let 0teA(A). I f a~(t), i = i , 2 , . . . is a sequence in 4 , and if

lim~ ~(t) = ~(t) e ~

(relative to the n o r m ] I on ~c~k) , then for all veV~, we have

limi X~(~(t))(v) = X~(~(t))(v),

in the sense that, for i sufficiently large,

z~,(~,(t)) (v) = z~(~(t)) (v).

24

THE A R I T H M E T I C T H E O R Y OF LOOP GROUPS 49

We note that Lemma (7. i6) implies that the action of g~(A) on V~ is continuous in the sense that

(8.5) If ~i, i-----~, 2, . . . , is a sequence in g~(A), if l im i~ i=~g~(A ) (relative to the norm [ [), and if v~Vk x, then for i sufficiently large,

~ . v = ~ . v .

Now let ~ A ( A ) , a(t)~_q~k, and choose a sequence ~i(t), i = I, 2 , . . . , in kit, t-l], such that lira i a~(t)=~(t) (relative to t ])- Then for ~fl~(A), wV~,

Ad Z~(~(t))(r:~(~)).v=lim~Ad Z~(,~(t))(~(~)).v, by (8.4') =limin~(~/~(a~(t))~).v, by (8. I)

--=r:~(lirni(~t~(a,(t))~)).v, by (8.5)

=r:kx(~t~(a(t))~).v, by (8.4),

and we thus have proved:

Lemma ( 8 . 6 ) . - /of a~A(A), ( ~ ( t ) ~ , and ~g~(A), then:

(8.7) Ad Z=(a(t))(rc~(~))= =kz(Y/=(a(t)) ~).

When there is no danger of confusion, we will write ~o for ~ . We then have ~ C g, = g.v(A).

For each eEA(A) and a(t)e~c~ o, we define an automorphism ~ of g.v(A) by

.~(a(t)) = exp(ad a(t)E~).

On the other hand, as we observed in Remark (5.8) (see (5. IO)) we have the central extensions (for R = k):

(8.8) a ) ~ ~ ~ o-+ Ok-+ g~(A) --> g~--> o,

b) o ~ c ~ g ~ ( A -~g.v(A ) )-+o.

Moreover, as we noted in Remark ( 5 . ~ ) , we have

~ ( A ) - - ~ ~ = ~ ( A ) * ~ ,

and ~, is just the projection onto the first factor. From this, one easily sees that

~o"~,(exp q ad ~(~)+~(~))= (exp q ad t~|

~eg~(.~), j~Z, a~A(A), q~k.

Then, from this, from (8.4) and its obvious analogue for ~L~(a(t)), and from the continuity of ~ , we obtain:

(8.9) r a~A(A), (~(t)e.~f, Beg~(.~).

225

7

5 ~ HOWARD GARLAND

Recall from w I, that g~(A) is said to be perfect if

(8. IO) [g~(~x), g~(~k)] = ~(~x).

Lemma (8 . ix) . - - I f • g~(A)---> g~(A) is a Lie algebra automorphism (regarding ~

gso(A) as a Lie algebra over k) , and i f g~(A) is perfect, then there is at most one Lie algebra c ~ automorphism • g~(A) -+ gk(A), such that

( 8 . x 2 ) ~ ' C0kO• = X o ~ k.

Proof. - - Assume x " : g~(A) ~ g~(A) is a second automorphism satisfying (8. I2) (with • in place of • Then set •215215 and note that

It then follows from Lemma (i .5) and our assumption that g~(,~) is perfect ((8. IO)), that • = identity. �9

We let Gaa,.~=G,a, so(A) CAut(gso(A)) denote the group of automorphisms generated by the automorphisms ~e~(,(t)), , eA(A) , , ( t)eSr We let

G,a(A ) C Aut(g~(A))

denote the subgroup generated by the automorphisms Y/~(~(t)), ~EA(A), ~( t )e~ ~ For any automorphism Y/eG~a(A), we have that Y/ maps ck into itself (and in fact

C e~ is the identity on ok, since ck is in the center of gi(A)), and hence ~ induces an automorphism ~e of gse(A); i.e., there is a unique automorphism ~ of g~(A) such that

(8.

In this way we obtain a homomorphism

gP': G~a(a ) -+ G~a,z(A),

where for ~/eG~a(A), the image .o~=gP'(~/) is defined by (8.I3). But then by (8.9) and the fact that (by definition of G~a(A)) the ~/~(a(t)) generate G~a(A), we have that qb' is the unique homomorphism from Gaa(A ) to G,a,.~(A), such that

r seA(A), ~(t) es

Then, by Lemma (8. I I'), r is injective. Thanks to the fact that (by definition of G~a, so(A)) the :Y'~(a(t)) generate G~a,.~(A), it is also surjective. Thus /f g~(A) is

perfect, we have:

Lemma ( 8 . x 4 ) . - - There is a unique group isomorphism

of Gaa(~ ) onto Gaa, se, such that

@'(Y/~(a(t))) = ~ (a ( t ) ) , asA(A), a(t)s~cP.

gO': Gaa(A ) --> Gaa,.~(A )

Remark. - - We have given the proof except to note that the uniqueness follows from the fact that the ~/~(a(t)) generate Gad(A ).

226

THE ARITHMETIC THEORY OF LOOP GROUPS 5I

9. S c h u r ' s L e m m a a n d a n i m p o r t a n t i d e n t i t y .

For the first part of this section we consider the special case when k = C. We fix X~D and, as specified at the beginning of w 7, we always assume that X(hi)Jeo for some i = I , . . . , t + i . From Proposition (6 . I I ) , we then have that

nx : g~(~) _~. End V x

is injective, and we therefore identify g~(A) with its image nX(g~(~)). In Defi- nition (7.2I), we introduced the group G = G ~ C A u t V x. We now prove:

Lemma (9. x ) (Schur's Lemma). - - I f g e E n d V x commutes with x, for each x~g~(A), then g is a scalar multiple of the identity.

Proof. - - In [7], w I2, we proved the existence of a positive-definite, Hermitian inner product { , } on V x, and of an involutive, conjugate-linear, anti-automorphism * of g(A) (where for xEg(A), we let x* denote the image of x under *) such that the weight spaces of V x are mutually orthogonal, such that

{x.vl, ( 9 . 2 )

and such that

( 9 . 3 ) u • (X)* = u (X).

Moreover, we saw in [7], w 12, that * has an extension to an involutive, conjugate- linear, anti-automorphism (again denoted by *) of ~(g(.~)). Again, if u~ / (g (A ) ) , we let u* denote the image of u under *

NOW,

( 9 . 4 ) VX ----~(fl(A)). v0,

since, e.g., V x is a quotient of V ~(x). Thus, to prove Lemma (9. i), it suffices to prove that g. v 0 is a scalar multiple of v 0. But for this, it suffices to prove that for any weight ~z.X of V x, and for v~eV~, we have

( 9 . 5 ) {g.vo,

However, we may write v~ as

V~=U.Vo,

where uE~//(u-(A)) is a linear combination of products of elements of u- (A) .

Thanks to (9.3), u* is a linear combination of products of elements of u+(A), and

hence

( 9 . 6 ) u*. v o = o.

227


But then

{g.Vo, v~}={g.Vo, u .vo}={u ' . (g .Vo) , v0}, by (9.~)

={g.(u*.v0) , v0} , since g commutes with g(A),

= o , by (9.6),

and this proves (9-5), and hence the Lemma. �9 In w 7 (formula (7.~o)), we showed that Z~(e(t)) is additive in ~(t). We shall

now use Lemma (9. i) to prove a second important identity for the Z~(~(t)). Thus, let e, ~eA(A), with e + ~ + o , and let e~=~i(t), i = 1 , ~ , be elements o f s where we now allow k to be an arbitrary field. For automorphisms A, B of V~, we let (A, B) denote the commutator ABA- ~B-~.

Lemma ( 9 . 7 ) . - We have

(9.8) (Z~,(~h), Z~(~rz))----- IIx,~,+j~(c,.~.].~), where the product on the right is taken over all roots i ~ + j } , i, j e Z , i , j > o , arranged in some fixed order, and the ci~'s are integers which depend on o~, ~, and the fixed order, but not on k, ~ , ~ . Furthermore cn satisfies

[g~, E~] = cl~E~+ ~;

i.e., the integer cn coincides with + ( r + I), in (4.1).

Proof. - - First, assume al, a2 are Laurent polynomials in t and t -~,

(9" 9) c~1 = Y'q~ tj

where the sums on the right arc finite, and the pjs and q~'s arc complex numbers. We let R = Z [ p ~ , qi], denote the ring obtained by adjoining the pj's and qj's to Z. Then VRxCV x, and the automorphisms X~(al) and X~(a~), defined in (7.19), leave VR x invariant.

Now thanks to (8.7) and Lemma ( 8 . I 4 ) , w e have

O' o ad(X~(a2))= ~e~(a2).

Also, thanks to [21], Lemma (15), page 22, we have that Lemma (9.7) holds for ~r ~f~(a2) in place of Z~(al), Z~(a2), respectively. It then follows from Lemma (8. i4) , that Lemma (9.7) holds for ~/,(al) , ~/~(~) in place of ?G(al), Z~(a2), respectively (note that g~(A) is perfect, since k = C ) . Then it follows from Lemma (9. i) that Lemma (9.7) holds for Z~(al) and Z~(a2) modulo scalars (given our special choice of al, as). Thus there is a complex numbers y=3 , (a l , ~2), such that

228

THE A R I T H M E T I C T H E O R Y OF LO O P GROUPS 53

where I denotes the identi ty operator. However, the left side of this equation leaves VR x invariant, and defines an automorphism of V x. It follows that y(%, %) is a unit of R. Now choose the p/s and q~'s to be algebraically independent over Z. Then we must have y(a,, a=)=4- i . Specializing the pi's and qj's to be zero, we find "r(a~, % ) = i. Again, by specializing, we see that in fact L e m m a (9.7) holds over any field k, provided a,, % are of the form (9.9), with the expressions on the right being finite sums. But then, thanks to (8.4'), we obtain Lemma (9.7). �9

IO. Relat ions wi th the work o f Matsumoto , Moore, and Steinberg.

In this section we wish to apply (7.20) and L e m m a (9.7) to show that G~ is a central extension of a classical Chevalley group. More precisely, if A is a classical Cartan matrix, we let G.~k=G~k(A ) be the abstract group generated by symbols ~ ' ( a ) , aeA(A), a e ~ so that one has the defining relations:

A) :ge:(z+v) = ag~'(a)~e:('~), aek (A) , ~, z e ~ ,

B) , ) = 11 ' , ; ,j>0

i~ +jG E A(A)

(where the order of the r ight-hand product , and the c 0 are as in Lemma (9.7)), provided A is not the I • I matrix A = ( 2 ) , in which case (B) is vacuous. But, if for any 0~eA(A) and non-zero aeg~ we set w '~(a)=~(a)~ '~( - -a-*)age~ ' (a ) , then for A = ( 2 ) , we have:

(7) = B') . . . .

and finally, if we set h'=(~)=w'=(~)w'=(I) -1, then

c )

Alternatively, if G denotes the Chevalley group scheme over Z, such that G c is the simply connected topological group corresponding to A, then G.~,k, defined above, is isomorphic to the group of ~ rational points of G (see [2o]). Moreover, if we define E(Ga% ) to be the group generated by objects Z~,(cr), ~eA(A), ~ e ~ , which, for X~(a) in place of .ge~(a), satisfy the relations A) and B) (or B'), if A = ( 2 ) ) , but not C), then clearly there is a unique homomorphism

such that q~,(Z;(~) ) = ~ge;(a), ~e.gPk, ~r

Moreover, Steinberg in [2o], shows that E(G.~k) is the universal covering, in the sense of Moore [18], of O2, k. We also have

Lemrna ( io . i ) . - - There is a unique homomorphism

----> G k ~

229

54 H O W A R D GARLAND

such that

�9 = zd ),

Proof. - - If A:#(2), then the Lemma follows from Lemma (9.7) and from (7.2o). We are thus left to prove the Lemma in the case when A = ( 2 ) . We let A 1 denote the 2 • classical Cartan matrix

a.(: :) and, as in w 3, we let El, E2, F1, F2, H1, H~ denote the generators of 9(A1). Similarly, we let E, F, H denote the generators of g(A). We then have an injection ~: g(A),--->g(A1) , defined by the conditions

~(E)----E1, t(F)---- F:, ~(H)=H1;

then ~ defines an injection

2 ~ : C[t, t-l] | t-1]|

where T = I | and I denotes the identity map of C[t, t- l] . In turn, 2" induces an injective Lie algebra homomorphism

Using the identification of g(A) (resp. of g(A1)) with the universal covering of g(A) =arC[t, t-1]| (resp. of ~(A1)=a~C[t, t-1]| given by Theorem (3. I4), we can give a simple, explicit description of ?. Thus let c>o be defined by:

c(X, Y ) = (X, Y)t, X, Y~g(A),

where ( , ) (resp. ( , )1) denotes the Killing form of g(A) (resp. of g(A1) ). As in w 2, we identify the universal covering of ~(A) (resp. of ~'(A1) ) with ~'(A)| (resp. with ~'(A1)| We then define "~ by:

( I0 .2) "~(~, d)=('~(~), cd), ~ ( A ) , deC,

One checks directly that

c gz(X1).

On the other hand, if XleD(A1), then X, the restriction of X 1 to I)"(A), is in D(A), and every element of D(A) can be obtained as such a restriction. As in w 6, we fix the "highest weight" vector v 0 in V h. It follows from a result of Kac (see [i I] and [8], Remark on page 6i) that ~/(g(A)).v 0 is the g(A) module V ~. Moreover, from [7], w167 II , I2, one sees that

V~ = V~i ("~ V K.

Thus, for any field h, we obtain an imbedding

:(A, A1) : Gk(i ) ~-->G~(A1).

930


To prove the Lemma for G~(A), we must prove relation B'). But this now follows from the existence of the imbedding ~(A, AI), from Lemma (zo. 1) applied to G2@~I) (which we have proved) and from Matsumoto [i3] , Lemma (5-i) (or Steinberg [~o]). �9

II~ Re la t ions w i t h the w o r k o f Bruhat and Tits .

We now consider the group E(Ga% ) defined in w io. Following Steinberg [2o], we can introduce a BN-pair structure on E(Gaek). However, here we also follow a suggestion of Tits and first introduce a donnde radicielle ([4], w 6.1) in E(G.~).

Thus, for each eeA(A), we let U~, denote the subgroup of all elements X~,(a), c r ~ . We set

w;(a)=X~(e)Z~_~(--e ~)Z~(~), ~eA(A), e e ~ - - { o } = ~ * ,

and we set he(o') = We~(O')W~(I) -1, eeA(A), ee~k*.

We let T e be the subgroup of E(Ga,k) generated by the elements hi(e), e e ~ - - { o } = ~ * . We let M~,, 0c~A(A), denote the right coset of T ~

M~---- T~w~(i).

oceA(A),

Proposition ( x x . x ) . - The system

(T *, (U~,, M~)~e~(A) )

is a donnde radicielle of type A(A), as defined in [4], w 6. I. Thus this system has the following properties (from [4], 6. i. i): (DR I) T * is a subgroup of E(G%), and for each ~eA(A), U~ is a subgroup

of E(Ga@, and U~, contains more than one element. (DR 2) For 0c, ~eA(A), the group of commutators (U~, U~) is contained in the

group generated by the U~+q~ for p, qeZ, p>o, q~o and p~+q~eA(A). (DR 4) For 0teA(A), M~ is a right coset of T *, and one has

ut -{z}c

where z eE(G%) denotes the identity. (DR 5 ) F o r 0c, ~EA(A), and neM~, one has

nU~n -1 _~ U~r~(~),

where r~(~)----~--2(0c, ~)(c~, ~)-1~. (DR 6) If U~ (resp. UL) denotes the group generated by the U~, 0ceA+(A)

(resp. A (A)) one has T"U~c~UL-----{I}.

Remarks. - - We have omitted (DR 3) of [4] since this property only applies to nonreduced root systems. Also, we note that the donn6e radicielle (T e, (U~,, M~)~ea(A) )

231


is generating in the sense of Bruhat and Tits; i.e., T" and the subgroups U~, generate E(G.vk). Indeed, the subgroups U~ generate E(Gsek).

Proof (of Proposition ( I I , I ) ) , - - (DR I) This is clear from the definitions. (DR 2) This follows from the definition of the U~, and from B), at the beginning of w io, as applied to the ;(~,(~) in place of the .~f'(a). Incidentally, though at first glance B) only applies to the case 0~4=--~, we note that (DR 2) is in fact a tautology if e = - - ~ . (DR4) The first assertion just comes from the definition of M~,. For the second assertion of (DR4) , we apply Matsumoto [i3] , Lemma (5.2) (h):

z -1 -1 (~ )w~(c~ ) ~ ( _ ~ - 1 ) , ~4:o, which follows from the definition of w;(a-1). (DR5) This is just a consequence of (7.2) and of (7-3), (c) in Steinberg [2o] (also, see Matsumoto [13] ) . (DR6) We define U~, 0c~A(A), to be the subgroup of G~k consisting of the elements .~e~(e), ~.Lfk" We let U+ (resp. U ) denote the group generated by the U~, 0:eA+(A) (resp. A (A)) and we let ~" be the subgroup of G.vk generated by the elements h'((~), 0:~A(A), (~es k. We then consider the homomorphism ~ : E(Gsok)-+ Gaok, defined in w IO. We have

q~(U~) =U~,, ee A(A),

= u •

v'(T ") = T .

Moreover, q0 ~ restricted to U ~_ is injective (see Steinberg [2o], (7.I) and Matsu- moto [I3] ). Hence, in order to prove (DR 6), it suffices to prove TU+ c~U ={I} , where I now denotes the identity in G.e, k. But this is an easy consequence of the representation theory of G.~ k (we can assume TU+ represented by upper, and l.l_ by strictly lower triangular matrices), and is well known. �9

For aeAw(A), we let

ro : - ,

(D~(A)*=dual space of I)~(A)) be defined by

ra(~) = ~--~(ha)a ,

where, as in w 6, after (6. I3) , we set

h a = 2ha/ (a, a), a Aw(A).

In Appendix I, at the end of this paper, we shall prove:

Lemma (xI .~ , ) . - - Let aeAw(,~), and let ~ be a weight o f V x, then:

, x such that (i) I f veVX~,k (see (6.2)), there exists v eV~,(~l,k,

Wa(S).V sek*.

(ii) ha(s), sek* acts diagonally on VX, k as multiplication by s ~(h").

(See (7.23) for the definition of wa(s), ha(s), with aeAw(A), sEk*.)

232

THE ARITHMETIC TIi,EORY OF LOOP GROUPS 57

Recall, that at the beginning ofw 7, we assumed X(hi)Jeo for s o m e i = I , . . . , t@I. It then follows from Lemma (iI .2) that w~i(i ) is not the identify automorphism of V x (just apply w~i(I) to a highest weight vector and use (i)). Hence G~ has more than one element. But G~ is generated by the elements )~(e), eeA(A), ae.Lf k (see Definition (7.2I)). Hence the homomorphism

^k

of Lemma ( io . i ) is surjective. Since GkX=#I, as we just noted, we have

(I x. 3) kernel tF~ 4: E (Gsek).

In G~ we let T ~ denote the subgroup generated by the elements h~(a), eeA(A), e e l * . We let U~ x, eeA(A) denote the subgroup consisting of the elements Z~(a), ae~cfk, and we let U~+ (resp. U~_) denote the subgroup generated by the U~ x, eeA+(A) (resp. A_(A)). The homomorphism tF": E(G~,k)~6x satisfied (and was uniquely determined by) the conditions

~Ve(Z~(o))----;G(a), eeL(A), cre.Lf k.

It follows that

(ix .4) �9

tF~ U ~ (

�9 F (T0 = T

Moreover, thanks to Proposition ( i i . i), and to Bruhat-Tits [4], Proposition (6. i. 12), we obtain a BN-pair in E(Ga,k) , given by the pair of subgroups (W, N~), where B~=T~U~_ and N ~ is the subgroup generated by the w~(~), pEA(A), ~e.Sf k. But then from Bourbaki [3], Theorem 5, P. 3% and from (II.3) , above, we have

( I I . 5 ) kernel W ' C Tq

For eeA(A), we let M~ x denote the right coset of T x

M~ =TXw~(I).

The following proposition is then essentially a corollary of our above remarks and of Proposition (i I. I):

Proposition ( i x . 6 ) . - The system

(T ~, (U~, x Ma) aeA(A))

is a donnde radicielle of type A(A).

P r o o f . - Properties (DR 2), (DRr and (DR 5) follow from the corresponding properties for (T e, (U~,, M~)~r and then applying the homomorphism tF*. For property (DR6), we note that if and g"eT ~U~_ such that

tF"{g') =tF~(g") =g.

geTXUX+c~UX_, then we may choose g'eU"_

233

8


Then, thanks to (I 1.5), we have

g ' = g " t , tekernel W ' C T ~.

But U~_nT"U~_={I}, and hence g ' = i . Therefore g = ~ e ( g ' ) = i , where, in this ^X

last equality, i denotes the identity element of G k . This proves (DR 6). It only remains to prove (DR I), and for this, it suffices to show that U~ x contains

more t hanone element. Indeed, if a ~ e k and a~eo, then Z~(a)+I. For if Z~(a)= i, then Z~,(a)~kernel~F ". But then, by (11.5) , we would have Z~(a)eT ", hence ~'(~)=q~"(Z~(cr))~T=q0~(T"), and this is only possible if ~ ~ (with I now denoting the identity in Gz,,k). This, in turn, implies e = o , a contradiction (see e.g. Steinberg [2I], Corollary I, p. ~6). �9

( I I . 7 ) . R e m a r k . - - It follows from the observations in the proof of Prop- osition ( I I .6) , that each of the maps

is injective.

We let v : ~qPk~Ru{ oo}

be the t-adic valuation. Thus, if

~ : Z ait% ~ , %+0 , i>go

then v(~) = io,

while v(o) = oo.

For each ~sA(A), we define functions

R u ( o o } , V~ : e__>

x U ~ ] R . w { or},

: oo}, = = by ~ ~ x ,

We let M~ denote the right coset of T,

M~ = Tw~,(i).

It is then known that the system

(T, (U~, M~)~e~(A) )

is a donn6e radicielle of type A(A) (see Bruhat-Tits [4], w 6. i .3) . We make the

notational convention that v~* (resp. U*, resp. M~) may denote any one of ,~*, v~x or ,J~ (resp. U~,, U~ x or U~, resp. M~, M~ x or M~). Similarly, we let T* denote either T e, T x

or T.

234


We now prove:

Lemma ( I I . 8 ) , - - The family (v*)~ea(A) is a valuation in the sense of Bruhat- Tits [4], w 6.2, of the donnde radicielle (T*, (U*, M~,)~eAIA)).

That is, the (v~,)~a(A) satisfy the following conditions (each ~* mapping U* into I (u{ oo}):

(Vo) For each ~eA(A), the image of v~, contains at least three elements.

(VI) For each ~eA(A) and ieRu{oo}, the set U~,,~=v~,-l([i, oo]) is a subgroup of U* and one has U~,,~={I}.

(V2) For each :~eA(A) and meMO,, the function

u~,7_~,(u)--,*~(mum -1)

is constant on U*_ ~ - { I }.

(V3) Let e, ~eA(A) and i, j E R ; if ~ r then the group of tators (U*,i , U~,j) is contained in the group generated by the

U;o: + q~,pi+ q~,

where p, q are strictly positive integers and p ~ + q ~ e A ( A ) .

(V5) I f a~A(A), ueU*, and u ' , u " e U * ~ , and if u ' u u " e M * , then one has

r r

commu-

Remark. - - We have omitted (V4) of [4] since that condition only applies to root systems which are not reduced.

Pro@ valuation (V~)~eA(A) , the Lemma is proved in Bruhat-Tits [4], (6.2.3) (b). consider the homomorphism

q~* : E(Gxk) ~ G ~ k �9

As we noted in the proof of Proposition ( i i . I), we have q)~(U~)=U~, and similarly, we have q~e(M~,)= M~. Moreover, we clearly have

( " . 9 ) =

- - For G.~ and the donn6e radicielle (T, (U~, M~)~eA(A)), with But then

q~*(Te)---- T,

Thus, (Vo) for E(G.vk) follows from the corresponding assertion for G.v k. By ( i i .7), q~' restricted to U~, is an isomorphism onto U~. Hence (VI) for E(Gq,k) follows from (I1.9) and the corresponding fact for Gxk. We now consider (V2) for E(G.vk). Thus, let m~M~; then the function

u~v*_ ~(u) - - v~(mum-1), ueU*__ ~--{ I },

is the same as the function

1),

235


which is constant as ~(u) varies over U_ ~--{ I }, and hence as u varies over U"__ ~--{ I }. Property (V3) for E(G~k ) follows from (B), at the beginning of w lO, but applied to the Z~(a) in place of the .~f'(a), ae s e~A(A). Finally, (V5) follows for E(G.vk) f rom the corresponding property for G.vk, and an argument analogous to the one used to prove (V2) for E(Gvk) (assuming (V2) for G~k ).

Finally, we come to the proof of L e m m a ( I I . 8 ) for the (v~)~e~(A)X and G k.̂ x Here we use the homomorphism ~ " : E ( G x k ) ~ G ~. Thanks to ( I I .4 ) and ( I i . 7 ) we have that for each aeA(A), ~ induces an isomorphism from U~, onto U~ x, where, recall,

�9 r~(z~(~)) = z~(~), ~ .

Since we clearly have

( I I . IO) "OXa(UieeCx)) = "oeaCx), xEUea,

we obtain (Vo) and (VI) for 1~ from the corresponding properties for E(G.vk). Property (V3) for G~ follows from (9.8). Property (V2) for ~x follows from (11.4) , (1 i . lO), from the observation, made above, that W e induces an isomorphism on U~ (see (11.7)), and from (V2) for E(Gse~). Similarly, (V5) for ~x follows from (V5) for E(G~k), f rom the facts just noted, and from (I1.5) , which implies

=M;.

x2. Computation of the symbol.

We fix XeD ( = t h e family of dominant integral linear functionals in b~(A) *, as in w 6). As specified at the beginning of w 7, we assume X(hi)#o , for some i = i, . . . , t + i. We consider the highest weight module V x, and fix a highest weight vector v0eVz x. We also let v 0 denote the element i |174 (where k continues to denote a field, not necessarily of characteristic o). As we observed in (6.7) , every weight ~ ot V x is of the form

~+1 ~ = X - - ~ n~ai, n~>o, n ~ Z .

We define dp(9) (the depth of ~) by r

dp (~) = ]~ n~ i=1

(see [7], proof of L e m m a (lO.4)). We let Vo, vl, v2, . . . , vm, . . . be a basis of V x consisting of vectors in the weight

subspaccs x x V~,k, ~ a weight of V x. We assume our basis ordered so that if vi~V~,k, x v~eV~,k, , and i < j , then dp (~ )<dp (~ ' ) . Moreover, for each weight ~ of V x, we

assume that the basis vectors in V~Xk appear consecutively. We will call such a basis coherently ordered. We note that, with respect to a coherently ordered basis, the

z36


elements of u + (resp. u~-, resp. I)k(A)) are represented by upper triangular matrices with zeroes on the diagonal (resp. by lower triangular matrices with zeroes on the diagonal, resp. by diagonal matrices).

We let .,a'uCJ denote the subgroup generated by the elements X~(a(t)), where either ~eA+(A), a(t)e0, or e~A_(A), a ( t )e~ . From the observations of the previous paragraph, and the definition of X~(a(t)) (see (7.19)) w e have:

(x~,.x) With respect to a coherently ordered basis, the elements of 5ru are represented by upper triangular matrices with ones on the diagonal.

Next, we have

Lemma

(~2.3)

q~ek, qo4:O,

( 1 2 . 2 ) . - - Let

a ( t ) = q o + q ~ t + . . . + q y + . . . ,

be an element of O*. Then, for ~eA(A), h~(a(t))

. h~(a(t))=xh~(qo), Xe~" U.

can be written as a product

Proof. - - We set

a(t) - a = q'o+q~t+.. . +q~tJ+. . . ,

where q~ek, qo*O, and in fact,

qo=qo ~. Also, we set

We then have:

(12.4)

p(t) = d t ) - - qo q(t)=a-l(t)--qo.

h, (a ( t ) )=w~(~( t ) )w~(I ) -1 (see (7.22)), =X~(a( t ) )X_~(- -a( t ) - l )Z~(a( t ) )w~(1) -1 (also by (7.22)), =z~(P(t))z~(qo)Z-~(-- q(t)) Z- ~(-- qol)z~(P(t))X~,(qo) w~,(1) -1 = Z~,(p(t)) Z_ ~(-- q(t))z~(%);G(qo)X-~(-- qol)X~(P(t))Z~,(qo) w~,(1) -1 =z~(p(t))z_~(--q(t))z~(qo)z~(p(t))u'/q~ by (7.22),

where for h, g in a group, we set gh=hgh -i . Also, a direct computation using (7.3), (7. I I), and the following consequence of (7. i i) already noted in w 7, preceeding (7.15):

7~ (s) = w x ~ , ( • s) (w ~) - 1,

shows that

( I2 .5) a d 7~(s)(n~(~))---- ]~ s"n~(ad(~/n!)(~)), n>_o

(finite sum),

But then, another computation, using (I2.5) , yields

Ad w~(q)(~(~a)) - ~ x = - q ~(~-o(~)+j3,

237

60

for

Similarly,

0) (ii)

where if


q e k * = k - - { o } , a=a(~t )+j~eA(A) , j 4 :o in Z. From this last equality, we have

- 2 t x~(P(t))t~176 P( ) ) ,

X_~(--q(t))'~ and it thus follows that

Z~(P( t) ) X- �9 (-- q( t) ) • (p( t) )wdqo)=

z~(p(t) ) (Z_~(--q(t) )x-~(q~)w-~(-q~))x~(p(t) )~(q,) ,

eeA+(A), the expression (i) is clearly in Jv , and if eeA_(A) , the expression (ii) is clearly in Ju . the expression (12.4) for h~((r(t)), we have

=zh (qo), and this proves Lemma (12.2). �9

Corollary. - - For (r(t) e ~*, with

~ ( t ) = q o + q i t + . . . + q f + . . . ,

and for ~ A ( A ) , we have

(x2. 6 ) h~( ~( t ) ) . v o = qXoCha(~))v o.

Pro@ - - We note that h~(qo)=h,(~j(qo) , L e m m a (i2.2), and (12.1). �9

We now wish to compute

h~(a(t) t-1)h~(t-1)-i h~(e(t) ) - l . v o,

Thanks to (12.6), it suffices to compute

h~(a(t)t-1)h~(t-1) -1 . v o.

Calling ;( the common value of (i) and (ii) and using

and apply Lemma ( I I . 2 ) ,

~eA+(A), a(t) e d?*.

But we write (r(t) as in (I2.3) , and then, as we noted in the proof of L e m m a ( i2 .2) , we have

a ( t ) - ~ = q o + f ~ t + . . . + q ~ t J + . . . ,

We write ~ for a(t), and recall that we defined p(t), q(t) by

= q0 +p( t ) ,

• - -1 = No I + q(t).

qo=qo ~, q'.ek. 3

(in the proof of Lemma (12.2))

We then have

where

(~t -1 =qot- l - l -~( t )

( ~ - l t = q o l t +'((t) ,

~(t) =p( t ) t - l eO ,

~( t )=q(t ) t~Ot 2.

238

We have

( I 2 . 7 )


w ~ ( . t - 1 ) = Z ~ ( . t - 1 ) Z _ ~ ( - - . - l t ) x ~ ( . t - t ) , by (7.22),

=z~(p(t) )z~(qot-~)Z_~(--~'(t) )z_~( - qolt);(~('f (t) )x~(qo t- i)

---- Z~ (Y(t))Z- ~(-- "q(t))x~(q~ t-~) ;G (P"(t))

= z~(p'(t)) Z_ ~(-- "~(t))x:r ).

Now of course

We wish to also show:

(recall 0~EA+(A)).

(I2.9)

where ~ = + I

(x2 .xo)

On the other hand, also by (I2.9),

z - ~ ( - ~(t) F-~') = z~ ( - ~ (a) -~ ~(t)),

(**. a) z- ~ ( - ~(t)) ~(~~ z,(Y(t))~'(~~ Jo.

However, thanks to Lemma (io. i), and to Matsumoto [I3] , Lemma (5-i) (b):

w,(~l) z~ , (~ ) w~(~,) -~ = z~ ~ ( w ? ~ ) ,

or - - I , ~i, a 2 ~ , ~ A ( A ) . From (I2.9) , it follows that

X~,(P(t)) w~'(q~ = Z-~,(• qo2t2p(t)) eJu .

63

cEk*,

'~ 2 where again, ~q---- + i or - - I . But, since q( t )eOt , cr we see that this last element is again in Ju . But, on the other hand, we have:

X_~(--~(t) )t~ )z~(c-l t -1)X_~(--ct) .

)_~(ct )eJu , we therefore obtain:

c ~k*,

Since

and setting c = - - qol, and recalling (I2. IO), we get (I2.8).

But then h~(a(t)t-1)h~(t-1)-l.Vo

= w=(a(t)t-i)w=(i)-iw=(i )w=(t-1) -1 . v0,

= w~(,(t)t-t)w~(t-1) -1. v o

= ~w~(qot-1)w~(t-1)-a'Vo, ~ J u ,

thanks to the expression (I2.7) for w~(a(t)t -1) and to (I2.8). But for

w~(ct -1) = z~(a -1) z - A - ~- i t )z~(a -1)

= z~(d z_o( - c-1) z~(c) = wo(~), a=a (~ ) -~ .

by (7.22),

c~k*

w~(qot-a)w~(t-l) -1 =: w~(qo)Wa(I) -1 -----ha(qo), and hence

h,(a(t) t -1)h~( t - , ) - l . Vo = ~h~(qo) . Vo = qXo(ha) ~ . Vo '

= qXo(ha)v o,

by Lemma (I 1.2)

Hence

239


since ~L~eJu, so we can apply (I~. i). Combining this last computation with (i~ .6), we obtain

h~(a(t)t-1)h~(t-~)-t h~(e(t))-l , v0 __-- qo x(a~(~))q~(a,)Vo --__ q~Vo '

2x(h:) a = a(~) -- ~ and co . . . .

where

f . 0 - -

(see w 6),

i.e., we have for :r

(x2. x I ) h~(~( t ) t -~)h~(t -1)-~hJ, ( t ) ) -~ , v o = q~vo,

We also note that

(I2. x2) h~(~)(qol)h~(qo), v o = q~v o.

On the other hand, if ~//(9k(A))----k| then

= k | = k | v0 = vo

2),(h:) (Z, (Z) "

and hence, thanks to Lemma (8.6) and to Lemma (8.11) we have from (12.11) and (12.12) (note that ha(~)(qol)ha(qo) =h~(qol)h~(qot-1)h~(t-1) -1)

(12. I3) h~(a(t) t-1)h~(t-1)-ih~(,( t)) -1 = ha(~)(qol)h~(qo)

=q~'I, ~ A + ( A ) , (~(t)----j~>oqjtJ , qo4:O,

where I denotes the identity operator on Vk x, and co is defined as in (12. II) .

We now consider the group G~ from the point of view taken in w io. Thus, we recall from w io, that the Z~(~) in place of the .oo'(~), ,r a~s satisfy the relations (A), (B) ((B') if A----(2)) at the beginning of w IO. But then, if we divide (~k x by the subgroup C generated by the elements h~((~l(~2)h~((~2)-lh~((~l) -1, ~eA(A), ai, *2eo~ (this subgroup is central by Lemma (IO. i) and by Steinberg [2o]), the resulting quotient group, G~k say, is also a quotient group of Gse k. Indeed, we have surjective group homomorphisms

f% x_+ ~_x 7~1 : " ' k " ' - W k ,

7~2 : ~----> GXk,

defined by the conditions

for ~eA(A), e e l . By Steinberg [~o], Theorem (3.2), we have that kernel(7:~) is ^ X

finite and central. We have already noted that C----kernel =i is central in G k .

240

THE A R I T H M E T I C T H E O R Y OF LOOP GROUPS 6 5

We let E~(G.~k)r215 k denote the subgroup of all elements (gt, g~), glEG~, g2eG~k, such that:

(~2. ~4) ~1(gl) =,~(g~).

Then, thanks to ( I2 . I4 ) , we have a commutat ive diagram

%

where q~t=qh x, ~ = ~ 2 x (we drop the superscript X when there is no danger of confusion) are induced by the projections of 0~• onto the first and second factors, respectively. We let ~---=~loq~l=rqoq~2; then since kernel(~l) and kernel(rq) are central in ~x and in G~k, respectively, we have

kernel(7:) C center (E x (G ~.k)).

We now fix a /0rig root eeA+(A), and we let

C x = kernel ~2.

Then, by Moore [I8], Lemma (8.4), the central extension

is determined by the function b~( , ) f rom s215 to C x, defined by

(**.'5) b~(~, ~) ' . . . . . . ~ ~ * = ~ - { o } ,

where for e~.o~k*

h~,' (a) = (h~(a), h'~(z)) eEX(G.~k).

We note that h~(~)h~(r ~, ~ ' ,

thanks to (C), at the beginning of w io. Hence, we have

(v,. x6) b~(~, .r)=(b.(cs, .r), i),

whcre

(~2.~7) b,(~, ~)=h~(~)h,(~)h,(~) -~, ~, ~'.

We Ict M denote an abelian group (in which we write the group operation

multiplicativcly) and wc Ict /~ denotc a ficld. Following Matsumoto [I3], wc let

S(k*, M) (k*=k--{o}) denote the group of all mappings

c : k*•

9


satisfying the following relations for

(x2. i8)

x,y, zd~*:

(Si) c(x,y)c(xy, z)=c(x, yz)c(y, z);

(S2) C(I, I ) = I ; c(x,y)-~c(x-l ,y-1);

($3) c(x,y)=c(x, (1- -x)y) , if X:~=I.

We let S~ *, M)CS(k*, M) denote the subgroup of those ceS(k*, M) which are bilinear; i.e., c(xy, z)=c(x, z)c(y, z), c(z, xy)=c(z, x)c(z,y), x,y, zek*.

Lemma ( I 2 . I 9 ) . - - Let eeA+(A) be a long root; then CX=kernel q% is generated by the elements b~(,, "~), (~, "~e~*, and we have b~( , )eS~163 *, Cx).

Proof. - - The first assertion follows from L e m m a (8.2) in Moore [I8]. When Gse k is not of symplectic type; i.e., when A is not the classical Cartan matrix corresponding to the symplectic group, then the second assertion follows from Matsu- moto [i3] , Theorem (5.1o). Also, Theorem (5.1o) of [13] asserts that in any case, b~( , ) e S ( ~ * , Cx). Thus, to complete the proof of the" lemma, we need only prove that b~( , ) is bilinear in the symplectic case. By (I2.16), this is equivalent to showing that b~( , ) is bilinear in the symplectic case (note that, also by (12. I6), we know that b~( , ) is bilinear in the non-symplectic case).

In the symplectic case we consider the subgroup I=I of Gk x generated by the elements )~(al(t)) , Z_~(e2(t)) , el(t), G~(t)e~f k. Passing to a suitable t2I subrepresentation of V~ (generated, over Z, by the highest weight vector and the subalgebra corresponding to the subgroup f-I), we may in fact assume that Gsek=SL2(~c~k) and that A is the i • i Cartan matr ix A =(2) . But then, arguing as in w IO, we may imbed gk(A) into g~(A1) where

is the classical Cartan matrix corresponding to SL 3. We may also assume V~ is obtained from a highest weight module for gk(A1), as the gk(.~)-submodule generated by a highest weight vector. We may then use the known bilinearity assertion for SLa, to obtain the desired assertion for SL 2. �9

We let

cT: x

denote the tame-symbol, defined by

(x2.2o) )Co)),

where we recall that ~ denotes the t-adic valuation on .W k. I t is known that cT( , )ES~ ) (see Milnor [I4] , Lemma ( I I , 5 ) , p. 98). Indeed, one can easily check bilinearity (which then implies (SI) of (I2. i8)) and property ($2) of (I~. 18),

242


directly. Property ($3) is also easily checked by noting, thanks to bilinearity, that it suffices to check c~(a, 1 - - ~ ) = i , for ae-~k*, a + I . One can in turn check this by setting ~ = a ' t " , a'eO*, and separately considering the cases re>o; re<o; r e = o , 6 ( O ) = I ; r e=O, 6(O)~=I.

Now we have observed that b~( , ) is bilinear (by (12.16) and L e m m a I2. I9), and thanks to (12.I3) , we know that

( io .ox ) b~(~r(t), t -1) is a non-zero scalar multiple of the identity operator on V~, for all ~(t)e0*.

In fact, we have:

Lemma (xo.o~,). - - For all ~1, %e~q~k *, b~(Crl, or2) iS a scalar multiple of the identity On V2.

for x ~ * ,

(s3')

But then

Proof. - - I f we write ! * ~i= ~ t ~i, ~ i e 0 , i = I , 2,

then we see f rom the bilinearity of b~( , ), from property ($2) of ( I2 . I8) , and from ( I2 .2I ) , that it suffices to prove the lemma for

(i) b~(%, %), %, % e & ,

(ii) b~(t, t-~), b~(t, t).

But, in case (i), each of the operators h~(%), h~(%), h~(%%) is upper triangular with respect to a coherently ordered basis (Lemma (12.2)). Then, we can use (12.6) to show

We note that since for ceS~ *, M),

x)=:c(I-., i - ( i - x ) ) ,

x=t=I, the relation (S3) of (I2. I8) implies

c( I -x ,x )=I , xeY' , x+I.

b~(t, t -~)=b~( t ( I - - t -1) , t-~), by ($3') and bilinearity,

= b~(t-- i, t -z) = non-zero scalar multiple of the identity, by ( I2 .2I ) .

On the other hand

b~(t, t )= b~(t, t~t -~) = b~(t, t2)b~(t, t-~), by bilinearity =b~(t, t)eb~(t, t-~), by bilinearity.

Hence b~(t, t )= b~(t, t-*) -1,

243


and hence b~(t, t) is also a scalar multiple of the identity. of L e m m a (I2.22). �9

By L e m m a (~2.22), we may identify b~( , ) By ( i2 . i3 ) , we have

~x(h:) b ~ ( ~ ( t ) , t - a ) = q o o~, r - -

(~, ~)"

for ~ ( t ) = q o q - q i t q - . . . +q~tJq - . . . , qoek *.

cT(a(t), t -1) = o(t)- l(o) = qo i,

and so b~(a(t), t -1)=c~(a(t) , t-~) '~,

This concludes the proof

with an element of S~

On the other hand,

~(t)~e'.

Similarly, 6T((Yl, 6 2 ) : I , for al, a~e0*, as one sees from the definition.

b.(.1,-~)=cT(-1, .,)~, -1, ~ r

But then, arguing exactly as in the proof of L e m m a (I2.22), we see:

Theorem ( I 2 . 2 4 ) . - I f r ~)-~, for ~eA+(A)

b~(~l,-,)=cT(~l, -~)~, ~,, ~ ' .

In particular, we may take X = k0, where

(x2.25) )'~ ~ i=l+I.i=I'''"t' Then o ) = - - I , and

ba(~l, o'2)=6T(o'1, o'2) -1.

Hence

We thus have:

Theorem ( x 2 . 2 6 ) . - I f XeD, and X(~)+o for some is a unique homomorphism

@x: EXo(G.@ _+ EX(G.@,

ox((z~o(a(t)), 2~(a(t)))=(Z~(a(t)) , -~e~'(a(t))),

Proof. - - For any X, we set

a long root, then:

such that

~e~(a(t)) = (Z~(a(t)), ~e~ (a(t))),

i = I , . . . , r then there

[3eA(A), or(t) ~ s

244

Now let ~/~'=~Xo(~e~o(a(t))) be a word

in EXo(G~k). To prove the theorem, it then suffices to show that the corresponding word ~tx in EX(G.~k), is equal to the identity.

z(t) e 5ek* , SeA(A).

in the ~o (a ( t ) ) , such that ~ tz~


Indeed, let Y/' be the corresponding word in the Z~(a(t)), in E(Gsek). We have the commutative diagram

% where for XeD (X(hi):~o , some i = I , . . . , t + I ) , we let px be defined by the conditions

px(Z;(~(t))) = ~x(e(t)) , ~eA(A), e(t) e~* ,

noting that px is well defined since the ~r (in place of ~e~(~(t))) satisfy the relations (A), (B) ((B') when Gsek=SL2(s of w IO. Then

and so ~t"ekernelq~ ". But then ~ is a product of elements h~(~)h~(,~)h~(a,~) -1, e, ve~* (and inverses of such elements) in E(G~k) (see Moore [I3] , Lemma (8.~)). Hence Nx~176 (resp. ~tx_=px(~,)) is the corresponding product of elements b~~ (and inverses of such elements) (resp. of elements b~(e, z) (and inverses of such elements)). But ~x~ i), where I denotes the identity operator on V~ z~ and I denotes the identity in G~k. It follows from Theorem (i~.24) , and from our choice of X0, that ~x is the identity in EX(Gz~). �9

13. The Tits s y s t e m .

We begin by summarizing some of the material in Bruhat-Tits [4], w 6. Recall that in Lemma ( i i .8) of the present paper, we proved that the family (V;)~A(AI is a valuation of the donn6e radicielle (T*, (U;,M;)~eA(A/). The elements of A(A) may be identified with elements of t)R(A)*, the real dual space of I)R(A). For each vetlR(A), for s~R~. ( = t h e set of non-zero, positive real numbers), and for each valuation (~ )~ (A) (so ~ = ~ ~, ~=x, or ~z~) of (T*, (U;, M;)~A(A)), we define a new valuation (~;)~eA(AI by

( I 3 . I ) +; (//) = S~; (//) -~- ~ (/)) ~EA(A), usU;.

We will denote a valuation (~;)~ea(A) simply by ~* (e.g. we will denote (v;)~ez~(A) by v*). We then write (13 .I) more simply as

(x3.x') Thus the additive group [jR(A) acts on the valuations of (T*, (U~,, M;)~e~CA/), via (I 3. I') (with s = I). We call two valuations in the same orbit of this IkR(A) action, gquipollentes.

245


On the other hand, consider the group N* generated by T* and by the M; ,

03.4) +;(,)-- eA(A), ueU .

As noted in Bruhat-Tits [4], (6 .2 .5) , ( I ) , o n e easily checks that

( I3 .5 ) /Z.(S[.L*-@/))=S(r/.~L*)-~vp(n)(/)), s e R ; , wbR(A), n~N*.

We let A denote the set of valuations dquipollentes to ~*. But of course we may identify A with IkR(A), and then furnish A with the inner product structure ( , ) on IkR(A), given by the Killing form. We note that if ~*EA, then it is natural to let tx*--v* denote the corresponding element of IkE(A ).

For ~IkR(A)* , and m~R, we let a~,~ (or *a~,~, if we wish to make the dependence on the valuation ,~* explicit) denote the half space

a=, ~ ----{xeA l , ( x - - v*) + m>o} .

We define the afine roots of & to be the half spaces %,m, ,~A(A), meZ, and we let X denote the set of affine roots. We let

6~ Y~•

denote the subset of all pairs (a, e), where a----a~,m, for some meZ. We let 0a~,~, the boundary of the half space a~,m, ~eA(A), meZ, be defined by

&=,~ ={x~A[ ~(x-- r + m = o}.

Recalling that A inherits the inner product ( , ) f rom t~R(A), we let

r~,,, : A - + A ,

denote the orthogonal reflection with respect to the hyperplane 0a~,~.

246

u~U~, ~eA(A).

of (T*, (U~, M~),~A(A)), and n~N*,

We note there exists a homomorphism

one has

where r~ is the reflection on I)~(A)* defined by

- - ~, Vt d)R(A)',

then, as noted in (13.2), we have

we define

~eA(A).

vp: N * ~ W ( A ) ,

such that for all 0~EA(A), nEN*,

( x 3 . 2 ) nU~n-1---- U~,

Moreover, we have Vp(M*)=r~,

( I3 .3 ) r~(~)---- Vt (~, ~)

(see Bruhat-Tits [4], w 6 .1 .2 (IO)).

I f neN*, and if w=*p(n)eW(A),

n-luneU;_l(:,l, for

Given a valuation ~*=(P~)~eAIA/ a new valuation n. ~* = +*, by


Quoting Proposition 6.2. I O in [4], we have:

Proposition (13.6). --- The space A is stable under the action of N* defined in (13.4). For n~N*, we let p(n) : A-+A denote the mapping p(n) : x-+n.x, x~A. Then p(n) is an automorphism of the Euclidean space A, and the canon#al image of p(n) in Aut(I)R(A)) (the group of linear automorphisms of t)R(A)) is ~p(n). Moreover:

(i) For each meZ, , cA(A) and n e M ~ = * �9 . -1 �9 , M~nU_~v~ (m)U_, , the mapping p(n) is the orthogonal reflection r~,,,.

(ii) For a=a~,m, ,EA(A), m~Z, we let U~=U*,m. Then, i f n~N*, one has

and

p(n) (a) ~x,

nU*~n- t _= U;(,)(a)"

We let T~ C N* denote the subgroup

T~ = p - l ( I ) ,

where I denotes the identity automorphism of A. We have:

Lemma ( 1 3 . 7 ) . - - T h e g r o u p To x, XeD, X(hi)4=o for some i = i , . . . , g + I , is the group generated by the elements h~(~(t)), eeA(A), ~r(t)~0*, and by the elements hat.l(S), sek*.

Pro@ - - Denote by ~x the group generated by the elements h~(e(t)), ~eA(A), ~(t)eO*, and the elements hat.t(s), sek'. Using the identity

(13 �9 8) ha/+l(s ) =/2)_ %(ts)w%(t) -1, sffk*,

(which may be checked directly from the definitions) we see that h~t;l(S ) EN x, sek*. Also, the relation

(I3" 9) w~ (or) Z~(v)w~ (~r) - t = Zr~(~)(:qcr-e'r),

~, ~A(A), ~, , ~ * , ~=~(~, ~)=•

c = 2 ( ~ , ~)(~, ~)-~, ~(~, -~)=~(~, ~),

(see Matsumoto [i3] , Lemma (5 . i ) , Steinberg [2o], (7.2), and apply our Lemma (io. I), implies the relation

(13. ~o) h~(,,)z&)h~(,,) -~ = z~(,, ~-~), d = 2(~,, ~)(~, ~)-~,

But (I 3.8) implies

hot .(s) = h_ %(st) h_ %(t)-~, s~k*,

oc, " r ~ * , oc, [3eA(A).

247


and (13.1o) then implies hat.x(S)eTX. On the other hand, (I3.IO) also clearly

impfies that the elements h~(,(t)) , eeA(A), , ( t )e0*, are in T~ x, and hence we have proved

Conversely, assume n~T~, so we have p ( n ) = I. Then ~p(n)= i, and hence (by Bruha t - t i t s [4], (fi. I . i i ) ) we have n~T x. Now T x is the subgroup of G~ generated by the elements h~(~(t)), ~ ( t ) ~ * , ~ A ( A ) . Thanks to (13.1o) and to Remark (I I. 7), for each ~ A ( A ) we may define a character ~ : u-~u ~, u~T x, u ~ *, on T x, by

( x a . x x )

Thus, if p ( u ) = I , then

~(u~a) = v(a), for all ~ e 5ek, aeA(A).

Taking ~r= I, it follows, still assuming p(u)----I, that

(I 3, 12) "r ~) = O, for all e tA(A) .

Now u may be expressed as a word in the h~(a)'s and their inverses (0~eA(A), aeooq'k*), u=u(h=(a)) . Let u '=u"(h; (a) ) be the corresponding word in the h~(a)'s and their inverses, in E(G~,). Then of course

�9 r (uO = u

-~Gk). Also, we let u eG~k denote the (see w IO for the definition of tF~: E(G.@ ^x corresponding word in the h'~(a)'s and their inverses. Then

(see w IO for the definition of q~' :E(G%)-+Ga, k). Now, thanks to s tandard results in the theory of Chevalley groups (see Steinberg [2o], (8.2)), u' is a product

u' = h'~,(ch) . . . h'~t(ot) , oi l&*, i = I, . . . , e.

Hence,

u' = h~(~l) . . . h~t(crt) mod kernel tF'.

But from Moore [i8], L e m m a (8.2), we have that if [~eA(A) is a fixed long root, then kernel ~F' is generated by the elements h~(o)h~('r)h~(cr'r) -~, ~r, "re~q'~*. But then

(x 3. x3) u = W'(u')=h=,(%) . . . h=t(et) mod A,

where A denotes the subgroup of ~x generated by the elements br ~), ~, ~e.g~*. On the one hand, each b~(o,-r)eAut V~ x, is a scalar multiple of the identity

operator of V~ x (Lemma (12.22)). Thus we have:

(x 3. x4) Every element of A = q~'(kernel W') is a scalar multiple of the identity operator of V x.

298


We let ul=h=,(a,).., h=t(~t). Then from ( I 3 . I I ) ,

( I 3 . 1 5 ) U s = ul,~ all gEA(A)

But if p(u)= I , (13. 12) and (13. IO) imply g

v( H ~<~',~i> ) ---- o, ~ A ( A ) , \ i~----'1 i

i.e., we have

Taking see that

t

Z <~,=~>~(~3=o, ~e~(A). i = l

73

(13. 13) and (13. I4) , w e have

<~, ~> = 2(=, ~ ) (~ , ~) - ' ;

0~ = %, . . . , 0% and using the fact that the Cartan matrix A is invertible, we

~(~3=o, i=1, . . . , t ;

2~9

10

therefore

UlE~ ~.

Thus, to prove L e m m a (13.7), it suffices to prove:

(x 3. 16) A C ~x.

Of course, (13.16) is equivalent to showing

(I3.16') b~(a, "r)eT x, a,-ce~*.

First, if ,e0*, then b~(a,t-1)e~ x, by (I2.13). I f % , % e & , we have b~(%, % ) = 1, by (12.23). We can then argue as in L e m m a ( i2.22) , to prove (I 3. I6'), and hence L e m m a (x 3- 7). �9

Let ~)CA be a bounded region; then following Bruhat-Tits [4], (6 .4 .2) , we define an integral valued function fn on A(A) by

fn(~) = i n f { m ~ Z [ a,,,~ ~ ~}.

We let

denote the subgroup generated by the union of the subgroups U~.X rn(~), ~ A ( A ) , and we set

), U_ha = U• = U x, c~Un.

Remark (t 3. I7) . - - Let C denote the subset of all x~A satisfying the conditions

~(x--~X)>o, i = 1, . . . , t ,

s(,-~)<I.

Then fc(~) = {o, ~eA+(A) I, ~eA_(A).


From Remark (13. 17) , we see that Uc is the subgroup generated by the elements Z~(,), ~ 0 , ~A+(A) , and by the elements Z~(a), ~ , 0~A_(A); i.e., we have

(x a. x8) ~% = Uc.

Also, if we let P/o denote the subgroup generated by U c and by T~, then we see that from (I3. I8) and Lemma (13.7),

(I3" X9) J = P/o"

We let W be the group of aft-me automorphisms of A generated by the orthogonal reflections r~,m, ~r m~Z. Thanks to Proposition (I3.6), we note that

W Cp(NX),

and we let ~ Z = p - l ( W ) . We let S C W be the set of reflections r~l,0 , . . . , r~t,0 , and r_ %,1; i.e., S is the set of reflections with respect to the walls of C. The relation (13. I9) , and the theorem of w 6.5 in Bruhat-Tits [4] now imply:

Theorem (x3.2o). - - We have N X n J = T ~ . The quadruplet ^x (Gk, j , ~X, S) /S a Tits system, where we identify W with NX/T~.

We recall that a quadruplet (G, B, N, S) is a Tits system if

(I3.2x) (I) B, N axe subgroups of G, B u N generates G, and B n N is a normal subgroup of N.

(2) S C N/(BnN) =d~W, consists of elements of order 2. (3) sBwCBwBuBswB, s~S, wEW. (4) For all s~S, we have sBsr

From Proposition (6.4.9) and Lemma (6.4.11) of Bruhat-Tits [4], and from the observation that (in the notation of Lemma (6.4.11) of [4]) ~to is empty, it follows that

(I 3. 22) J = U_,cT~U+,c.

We also have from Proposition (6.4.9) of [4], that if one is given an order on A+(A) and forms the product

H G A• Ua'/c(~)

with respect to this order, then

(x 3. o3) The product map ~el-I(a)U~X,/~(~)-+U+,c,~• is bijective.

14. The Tits system (continued).

Our first goal is to define a certain subgroup of N~. show:

Toward this end we first

260


Lemma (x4.x). - - For mtZ, etA(A), we have

p(w~(tms)) = r~,m,

for all stk*.

Proof. ~ Let wI)(A) and set

+ ~ = ~ + v

(see (13 .I) and (13 .i ' )) . We then have, for n tN x,

n.~bX=n..oX+"p(n)(v) (by (I3.5)).

But, by (13.2) and (i3.9) ,

(x4. i ' ) *p(w~(t~s))::r,, e tA(a) , meZ, sek*.

In other words (see (I3.3))

(14.2) ~p(w~(es))(v)=v--e(v)H~, , w b ( a ).

On the other hand, by (I3.4), and (I4.I ' ) , with

(~4. s) (~. ~)~(u) = ~(~l(w~(t~s)- ' ~w~(t~s)),

But (I 3.9) implies:

(x4.4) w=( . ) - a Z~(v ) w=(.) = X,=(~l(~-*v),

with ,eoW~', veNCk, c---=2(~,~)(~,~) -1, ~ = ~ I .

w~(~) z~(~)w~(~) -~= z,~(~(~-~

hence

n = w~(t"s),

~ta(A), ,~U~. we have

Indeed, by (I3.9) , we have

~ ' , ~ ,

w~(~)-lz~,(,')w~(~) = z,~(~,)(~~

where ~'=r~(~), z '=~e-%-. But

c=2(~, r~(~'))(~, ~ ) - 1 = - 2 ( ~ , ~')(~, ~)-',

and we thus obtain (I4.4). But from (I4.3) and (I4.4) we have

(x4.5) w~(tms).,~x--,~X--mH~, ~tA(A), m~Z, s~k*.

Hence, for n = w~(tms),

n. +x = n. v x + ~p(n) (v) =,~x mH~ + (v-- ~(v)H~),

by (14.2) and transformation

75

(14.5). To prove Lemma (i4. i), it thus suffices to show that the

v ~ v - - m H , - - ~(v)H~

is r~,,,, the orthogonal reflection with respect to the hyperplane ~(v)=--m. is clear. �9

But this

251


Now let aCAw(A), and assume

a = a ( e ) + m , eeA(A), neZ.

(See (4-3) and the subsequent definition.) Then

w~(s)=za(s ) z_~( - - s -1 )7~(s ) , by (7.23)

= ) ~ ( t n s ) z _ ~ ( - - t - " s - 1 ) z ~ ( t " s ) , by (7. I9)

=w~(t"s) , by (7.22).

we have

S Ek*,

In particular, for at + 1 = - - % + ~,

w o , . ( s ) = w %(ts),

while, for i---- I, . . . , t,

(x 4. 6') wa,(s ) = w,i(s), a i = a(o~i) , sek*.

It thus follows from Lemma (i 4. 1) that, for sek*,

[ r~i,o , i = I , . . . , [ (*4.7) p(wo,(s) ) =

tr_%,l, i = t + I .

In fact, thanks to (I4.6), (I4.6'), and (I4.7) , we have proved:

Proposition (x4.8). - - Each of the elements Wa(S), acAw(A), sck*, is in ~I x, and

/of N C~x is the subgroup generated by the w,(s), aeAw(A), sek*, then under p, the

set {wai(x)}i= 1 . . . . . t+l maps b~jectively onto S. In particular W=p(N) .

The next lemma follows from the axioms (i 3 . 2I) for a Tits system and from the Bruhat decomposition (see Bourbaki [3], Chapter 4, w 2.3, p. 25).

Lemma (I4.9). - - Let (G, B, N', S) be a Tits system, and let NoCN' be a subgroup such that the projection

rc : N ' - + N ' / ( N ' n B ) = W

maps No onto W . We then have: (i) No nB inclusion N0,--~N' induces an isomorphism from

tify No/(NonB)

is a normal subgroup of N O . (ii) The

N0/(N0nB) onto W. (iii) I f we iden- with W , by means of this isomorphism, then (G, B, No, S) is a Tits system.

Proof. - - Since n(N0)=W, it follows from the Bruhat decomposition (Bour- baki [3], Chapter 4, w 2.3, p. 25)

G = [J BwB, wGW

that BtJN 0 generates G. The other axioms for (G, B, No, S) follow directly from the corresponding axioms for (G, B, N', S). []

From Proposition (14.8), Lemma (14.9) , and Theorem (13.2o), we have

252


Theorem (I4. IO ). - - The quadruplet

J , N, S)

is a Tits system.

We now wish to study the intersection N n J . Toward this end, we first study J . Thus, we let x H~ = H k denote the subgroup of G~ generated by the elements h~(s), eEA+(A), sek* and by the elements hat+~(s), sek*. From (14.6) we have

(x 4 . I I ) hat+l(s ) = Wot+~(s)w,t+~(I)-i =h_%(ts)h_%(t) -x, sek*.

But then, from (14. I I) and (13. IO) we deduce

Lemma (I 4. I2). - - The subgroup H k normalizes the subgroup Ju.

On the other hand, Lemma ( I2 .2) implies that J u and H k generate J , and so

Also, since h~(s)=hai~)(s), eeA(A), sEk*, we see by Lemma ( I i . 2 ) , (ii), that:

(x4.x4) Relative to a coherently ordered basis of V~, the elements of H k are represented by diagonal matrices, which restricted to a weight space V~,k, are scalar multiples of the identity.

Also, by (12. I) we have that relative to a coherently ordered basis, the elements of ~'v are represented by upper triangular matrices with ones on the diagonal. Indeed, the argument leading to ( i 2 . i ) in fact shows:

(x4. x5) Relative to a coherently ordered basis, the elements of .~'u are represented by upper triangular matrices with ones on the diagonal. Moreover, the diagonal blocks corresponding to weight spaces, are identity matrices.

We will use this stronger conclusion a little later on.

and (14 . 14) we see that

(x4. I6 ) Hkn J~-----{ I } ,

and hence we have:

But even from (I 2. I)

Proposition (I 4. x7). --- ~" is the semi-direct product of H k and the normal subgroup Ju.

Corollary (x 4. x8). - - The group H~ is precisely the set of all diagonal elements (with respect to a coherently ordered basis of V~) in J . Relative to a coherently ordered basis of V~, the group J consists of upper triangular matrices whose diagonal blocks corresponding to the weight spaces, are equal to scalar matrices.

We are now ready to prove:

283


Proposition (x4.x9). - - The intersection N n J is equal to H~.

Proof. - - Thanks to L e m m a (I I . 2), the elements of N permute the weight spaces of Vk x. Thus, by Corollary (14. 18), the elements of N n J must have off diagonal blocks zero, hence must be diagonal. Since H k C N c ~ J , the L e m m a now follows from Corollary (i4. I8). �9

We conclude this section with:

Lemma ( I 4 . 2 0 ) . - - The group H k is the group generated by the elements h,i(s), i = i, . . . , l + i, s~k*. Moreover, t-I k contains h~(s) for every a~Aw(A), sek*.

The proof of this lemma depends on:

Lemma ( I4 .2 I ) . - - For every a~Aw(A), the coroot ~a/a(a, a) is an integral linear combination of the fundamental coroots 2ai/~(al, a~), i = 1, . . . , g -}- i.

The proof of Lemma ( I 4 . 2 I ) is exactly the same as the proof of L e m m a (4.4) in [7] and is therefore omitted.

Proof of Lemma (14.2o). - - By definition, the group H k is generated by the elements ha(~)(s)=h~(s), 0teA(A), s~k*, and by the elements h,t+l(s), sek*. But

a(~) =a~, i = I, . . . , t, and hence H~ contains the elements h~i(s), i = i, . . . , t + i, sek*. The rest follows from L e m m a ( i4 .2 i ) , and from Lemma ( I I .2 ) , (ii). �9

15. C o m p a r i s o n o f the Cak x for d i f ferent X.

As always since w 7, we let k denote a field of arbitrary characteristic, and XaD an element such that X(hi)4:o for some i = I , . . . , g + I . From now on we call such an element of D normal. Then for each normal XaD we have defined the group GkXCAutV~. For each i = I , . . . , t + I , we let k~a~)*(A)* denote the dominant, integral linear functional (i.e., element of D), defined by the conditions

(x5 . i ) Xi(h~) = ~0, )~(n) = o , i , j - - - - I , . . . , l + i .

By restriction we may regard each X i as an element of t)(A)*, the dual space of t)(.~). We let EC t)(A)* denote the Z-lattice of rank g + I , spanned by ;h , . . . , Xt+ 1. Also, by restriction, each a~A(X) defines an element of ~)(X)*, and we let ErC I)(A)* denote the Z-span of the restrictions of elements of A(~). Then ~r is the free Z-module of rank g, generated by a l , . . . , at, restricted to I)(A), and since ai(h)~Z , i , j = I , . . . , ~ + I , we have E rCE For each normal, dominant integral ~eI)*(A)*, we let .~xC ~)(A)* denote the Z-span of the restrictions to I)(.~) of the weights of V x. F rom our previous observation that .~rC E, we immediately deduce that .=xC .~ In fact, we have:

254

in E


Lemma ( I5 .2) . - - Let k e D be normal. Then E , C ~ x C ~ , and ~x is of finite index

Proof. - - We have already noted that '~x C E. Also, since over Z, F,r has rank t, we note that if .~,CEx, then '~x must have rank t-}-i over Z, because X restricted to t)(A) is clearly linearly independent from the elements of ~.~. Hence, since .~. has rank g + i , it suffices, in order to prove the lemma, to show that .~C.~x.

We let I = { i e { i , . . . , t+i}lx(hi)4=o }. By our assumption that X is normal, I is not equal to the empty set. Let voEV z be a highest weight vector. We then have for i e I

Thus f / . v o 4: o,

(~5 .3 )

the difference each i~I). Now let j e { i , . . . , t + i } - - I . in { I , . . . , t + I } , so that

j : I , L =J,

and

(~5.4) ~(%, % , ) . o , m=~ , . . . , s - i .

We assume we have chosen

(I5-5) X(h;,) =o ,

We define vqeV x, q = I , ..

Dq ~fjq. Yq_ i,

e,. (f, . Vo) = A . (e, ~0) + h,. % = X(h~)~0" o.

and since, as one checks directly,

ga. V~C V ~ + , , x x aeA(A) and ~, ~ + a weights of V x,

X--a i is a weight of V x, and hence a i restricted to t)(A) is in E x (for Then we can find a sequence J l , . . . , J ,

the above sequence j l, . . . , J s so that s is minimal.

p ~ 2 , . . . , S.

., s, inductively by

Then

255

where v o is the fixed highest weight vector. We set ? ,q=X--ah- - . . . - -a jq . I f vq4:o, then v,4:o and vr~V~r , for r - - l , . . . , q . Therefore ah, . . . ,a jq restricted to I)(2~) are all in ~x- Thus, if we make the inductive hypothesis:

(H) for q with I < q < s , we have vq4=o;

then to prove the lemma, it will suffice to show that vq+10eo (note that we have verified (I-I) for q = I ) . But

0 5 . 6 ) %+1" vq = o,

thanks to (6.7), (15.3) and our minimality assumption on s (the last implying aiq+l+a,,, I < re<q) . Hence

e~q+l- 7Jq + 1 :ejq+l'fJq+l'V q =hjq+l.vq, by (I5.6)

= x,(h~.l)v,,

8o H O W A R D G A R L A N D

where the last equality follows from our previous observation that if vq+o, then vqeV~q. But our minimality assumption on s implies

r , a~q.~) = o , I <re<q,

a n d = o .

Hence, ~kq(hjq+l) x % ( h j q + l ) ,

and the latter is not equal to zero, by (I5.4). Hence %+.vq+~:~o, and vq+~ is different from zero. We have already noted that this proves the lemma. �9

Lemma (i 5. 7). - - I f XED is normal, then for each normal element ~eD, there is a positive integer m such that .~,,~C E x.

Proof. - - Since both E x and E~ are of finite index in E (by Lemma ( 1 5 , 2 ) ) , '~'t~ is in the Q-span of Ex, and hence m~eE x for some positive integer m. But E, CEx (by Lemma ( i5.2)) ; therefore Em~CEx, by (6.7) (as applied to t~ in place of X, and to a weight ~' of V ~, in place of the "~" appearing in (6.7)). �9

Our next point is to note that:

Lemma (x5.8). - - An element ~ of the center of G~ k is a product t S =h;,(s,) . . . h ~ v ( t ) , s~, ..,st~k*.

Proof. - - By Steinberg [2I], Lemma 28 (d), p. 43, we have

~ = h ; l ( ~ ) . . . h ; / ~ t ) , ~ , . . . , ~ t ~ ' ,

and t

(*) 1-[ *~J'~i '= 1, j = 1, . . . , t,

where (~, ~ ) = d # ( ~ , ~)(~, ~)-1, ~, ~eA(A). to both sides of ( . ) , we have

t

Hence, since the Cartan matrix A is nonsingular, we have and we may set s~= a~ek*. �9

We are now ready to prove:

But then, applying the t-adic valuation v

~(-3 = o, i = 1 . . . . ,e ,

Theorem ( I5 .9) . - - Let Xl, ?,~ be two normal elements of D, and assume Then there is a unique group homomorphism

~-~ ;k 1 ~'~ ;k 2 ~(xl , ~ ) : t-k -+~,~ ,

^ )'2 of 0~1 onto Gk, such that =(X,, ~)(X~l(cr(t))) = X~'(6(t)), ~EA(A), or(t) ~.s k.

Ex~ C E h.

256

THE ARITHMETIC THEORY OF LOOP GROUPS 8i

Proof. - - Let

w ~1 = w(z~,(6(t)))

be a word in the X~l(6(t)), 0~eA(A), 6 ( t )e~ ,* and assume w h : i in Gk .^zt Let w x~ (resp. we; resp. w') denote the corresponding word in the Z~2(6(t)) in G~* (resp. in the Z~,(6(t)) in E(G~k); resp. in the . ~ ( 6 ( t ) ) in G.~k). We consider the diagram

k

% where =1, ~2 are as in w 12. We have

= ~l(w~,) = ~2(w'),

and hence w' is in the center of G~k. But we also have the homomorphism

o f w IO, where Ve(Ze~(6( t ) ) )=.qf~(a( t ) ) , 0~eA(A), 6(t)e~k* , and hence

~0~ ~) = w'.

Then, by Lemma (i5.8) , there exist s~, . . . , st~k* , so that

. , ~ S (x 5. xo) we =h~l(sl) ' h~t ( t ) mod(kernel q0~).

I f we fix a long root ~eA(A), then kernel q~ is generated by the elements

(~5. v,) ~(,,~, 6~)= h'~(,,~)h~(,,~)h'~(6~,,~) -~, ~, 6 ~ * ,

thanks to Moore [ I8 ] , Lemma (8.2). ^X

In Lemma (IO. I), we defined the homomorphism W ' : E ( G ~ ) - + G k , such that

( '5 . i 2 ) v ' (z~(~( t ) ) ) = z~(6(t)),

~eA(A), ~ ( t )e~* . We write tF~'X for iF', to denote the k-dependence. We then have from (15 . 12) that

xi ~ ~i -1, a ~ * , i = I, 2, �9 r ' , ~ ; (~ (~ , ~ ) ) = h ~ (6~)h~ (~)h~ (~6~) ~,

and, thanks to Theorem (I2.24),

( I 5 . I 3 ) tre'h'(Ca(61, 62) ) =CT(61, 62)~(k/)I, G1, ff2e~k *, i = I, 2,

where r162 ~) -1 i : i , 2, and I denotes the identity operator on V~.

On the other hand, by (I2.13) (second equality)

(I5. I4) hXa(~)(c-1)hXa(~)_t(c)'=c~(xi)l;

i.e., taking into account (15. I3) , we have

~ ) )ho(~)_#~(~, 6~)).

267 11


Now we use (I 5. IO) to write w e as a product �9 . he t'S ~ e we=h~,(sl) �9 ~t~ t J2",

where pe is a product of elements c~(al, a2) , as defined in ( i 5 . I i ) . We let pZi i = I , 2, be the corresponding product of elements

,

in G~. We then have, from (I 5. i5) ,

W" ~(p,) =pX~, i = i, 2.

Hence wZi=~e'?~'(We)=h~(S1)... ]ZaZff(s/)p Xi, i = I , 2,

In other words, we have written w z~ as a product in elements h~(s~), aEAw(A), s~k*, and w x' as the exactly corresponding product in elements h~'(s~), a~Aw(A), s~k*. Thus, if wZ~=I, as we assumed, then wZ '= i , thanks to L e m m a ( I I . 2 ) and our assumption that Ex, C Ez .

I t follows that if we set

r~(X 1, Xz)(Z~'(a(t)))=Z~'(a(t)), ~eA(a ) , ~r(t)e~*,

then ~(Xl, X~.) defines a homomorph ism from G~ onto G k . �9 We conclude this section with a few observations. First, note that in the proof

of Theorem ( i5 .9) , we showed:

(x5 . I6 ) The kernel of the homomorphism

r~l : G~ ~G~k

is contained in H k. I t then follows from (I 5.16) and from (I4.16), that

^~ (x5 . I7 ) The homomorph ism 7q, restricted to the subgroup or is

injective. Also, we have, using L e m m a (i 5.8):

(x 5. i8) The kernel of the homomorph ism

re2 : G.%--~ G~ k

is contained in the subgroup H~CGsek, generated by the elements h~(s), ~ A ( A ) , s q k * .

Indeed, (i5.I8) results from Lemma (I5.8), and from the fact (see Stein- berg [2I], Corollary 5, after L e m m a 28, page 44) that kerne lT~CcenterG~k.

Finally, we note:

(x5.x9) I f ~z CEx,, for ~1, ~2 normal elements in D, then

( .) k S =mX 1 q- integral linear combinat ion of al, . . . , a t + 1,

where m is a strictly positive integer, and one has mXleEz.

258

THE A R I T H M E T I C THEORY OF LOOP GROUPS 83

Proof. - - Clearly X 2 may be expressed as in (*) with m an integer, so we need only check m > o (mXleEx, follows from the fact that E1CEx,, by k e m m a (I5.~)). Evaluating both sides of ( . ) on h~, we have:

= mXl(h3 , t + l

since a(h~)=o, for all a~A(X). But h~=~=tq~hi, qi>o, and since Xx,)t~eD are normal, we must have m>o.

I6. The Iwasawa decomposition.

With the exception of Lemma (i6.3) , in this section we take k = 1 1 or 13. Now it follows from w I2 in [7] (see the discussion following (12.6) in [7]) that

(x6. x) 4;= 4_o, a Aw(X).

As a result, * leaves 0R(A) c gc(X ) invariant, and hence we may regard * as an 11-linear, involutive, antiautomorphism on gk(X), k----either 11 or t3. Of course, if k = 13, then * is conjugate-linear.

As we discussed in w 9 of the present paper, we also obtain from [7], w I~, that there exists a positive-definite, Hermitian inner product { , } on V~ (k=11 or 13) such that we have (by (9.2) and ( i6 . i ) ) :

(I6. It) For all aeAw(X), the element ~a of gk(X), regarded as an operator on V~ x, is the Hermit ian conjugate of ~_a, with respect to { , }.

We thus let * either denote the Hermitian conjugate with respect to { , }, or the involutive, anti-automorphism of g,(X) introduced in w 9 (also see [7], w I2), and defined on gR(X) by restriction from go(X). From (16. I) we then have

(x6.2) 7~(s)*= X_,(s), s~k , a~Aw(A ).

Remark. - - In particular, )~(s)* is defined!

We now prove:

Lemma ( ,6 .3 ) . - - Let k be an arbitrary field. For each aeAw(A), we can define a

homomorphism

%:

by the conditions

o((o tF I = a((s

259


We also have

06.4) �9 o ( (_ i o

qek*.

Proof. - - Once we prove the existence of ~a , the identities of (16.4) will follow by direct computat ion. In view of the well known presentation of Sl~.(k) (see Stein- berg [21], Theorem 8, w 6) it suffices to check:

a) Z• is additive in s (s~k); b) ha(s ) is multiplicative in s (ssk*); C ) ' ~ - - 2 " "~' Wa(S)7~(S)Wa(--s)=z_~(--S s), S~k*, s~k.

However, we observed a) in (7. I5), and b) is an immediate consequence of (ii), of L e m m a ( I I . 2 ) . As for c), we first note that if a = a ( , ) + m , eeA(A), nEZ, then

Za(~)= "~ z~(t s ), Wa(S) = wAr"s).

Also, a) and a direct computat ion, imply that

Wa(--S ) = Wa(S) -1. Thus

Wo(S) Za(~)Wa(--S)= wAt"s) z~(t"~)w~(t"s)-I

where ~ = + i . this case that

We again

2 ~ = z _ ~ ( n t - " s - s ),

= z - a('~s- 27), by (I2.9) ,

Indeed, by Matsumoto [13] , L e m m a (5. I), b) and c), we have in :q = - - i . �9

assume k = R or t3, and for

denote the conjugate transpose. We set

and note that, thanks to ( I6 .e) ,

% ( z ~ ( s ) ' ) = % ( z • s~k.

Since the Z+(s) generate Sly(k), it follows that

(x6.5) for each aeAw(A), we have Wa(g)*=tF~(g*), geS12(k).

260

THE ARITHMETIC THEORY OF LOOP GROUPS 8 5

But then, thanks to (I6.4) , we have from ( i6 .5)

( I 6 . 6 ) w~(I)* = wa(I) -1 , a~Aw(X ) .

Definition ( i 6 . 7 ) . - - We let ~ = K : x C G ~ denote the subgroup

~:={g~(~Zlg. is defined, and equals g- l } .

Theorem (x6.8) (Iwasawa decomposition). ~ We have

6 =F:J.

The proof of this theorem rests on the following lemma for Tits systems, from [2I]:

Lemma ( x 6 . 9 ) . - - Let (G, B, N, S) be a Tits system, and let S =(r~)~ei, with I a suitable index set. For each ie I , let Yi be a set of representatives for the family of cosets

Br, B/B.

/ f w e W = N / ( N n B ) , and i f

w = 5 . . . , r~,

is an expression of minimal length for w in terms of the rj, then

( I6 . IO) BwB = Y h . . . Yjm B.

Proof. - - We make full use of the standard results on Tits systems (see Bour- baki [3], Chapter IV), and we argue by induction on m. By assumption, there is nothing to prove for m = I . Assume we have proved (I6. io) for some m ~ I , and let

W = r ~ l . . . ~m+l

be an expression of minimal length; then

BwB : B r h . . . r~mBr~,,§

= Ys , . �9 �9 Yjm(Br m+l B)

: Yh"" Y~'~YJ~.I B,

where the first equality follows from the minimality of m q-i and a standard property of Tits systems, where the second equality follows from our induction hypothesis, and where the last equality follows from our initial assumption concerning the Y~. �9

We now prove Theorem ( i6 .8) . - - We let KCS12(k) denote the subgroup

' s v ( 2 ) , k = c

K = SO(2), k = R ,

261

86 HOWARD GARLAND

and we let B1CSI~(k ) denote the subgroup of upper triangular matrices. For each aEAw(A)nA+(A), we then have

(IS.IX) tFa(n ) C ~

~Fa(Bx) C J ,

where the first inclusion follows from (16.5), and the second inclusion, from (16.4) and the definition of ~F a.

We now apply Lemma (i6.9) to the Tits system

(6~, J , N, 8)

of Theorem (x 4 . Io) , where we take our index set I to be { i , . . . , t + i } . First, by

Proposition (I4.8), we have Wa,(I)eN x, i = I , . . . , t + I ; therefore each wai(i), i = i , . . . , t + l , normalizes T~, since T~ is, by definition, the kernel of the homomorphism

p : ~ W ,

of w167 13-14 . But then, from this, and from (13.22) and (I3.23) , we have

(I6. X2) ~Wai( I )JS---~ UaiWai( I ) , f ,

where Uai C ~x is the subgroup

Uai={Zai(s)}se~,, i = I , . . . , t-~-I

(we note that (16.12) holds for an arbitrary field k). Then, thanks to the Iwasawa decomposition

S12(k) = KBI,

to (I6.4) , (16. i i), and (I6.12), we may take the Yi of Lemma (16.9) to be contained in q~,i(K), i = I , . . . , g-~-I. Hence we obtain Theorem (16.8) from Lemma (x6.9) , and the Bruhat decomposition for a Tits system (see [3], Chapter IV, w 2, Theorem x). �9

Next, we obtain a uniqueness result concerning our Iwasawa decomposition. Thus, we let HHk, + denote the subgroup of I-I k generated by all h~i(s), where s is a positive real number, and we let HHk, 0 denote the subgroup of H k generated by all h~i(s), where s has absolute value one. It follows from Lemma (x4.2o) that the elements hal(S), sek*, i = I , . . . , g - t - l , generate I-I k. We then have

(I6. I3) Hk = HHk,0Hk,+,

and all the elements of H~,+ (resp. of Hk,0) have eigenvalues, when considered as automorphisms of V~, which are real and positive (resp. of modulus one), thanks to Lemma (11.2). Since the elements of t-Ik, 0 act on each weight space as a scalar

operator of modulus one, and since the various weight spaces are orthogonal with res-

pect to { , }, we have that Ilk, 0 C ~ , and

GkX = KHk,+ Jv .

262

T H E A R I T H M E T I C T H E O R Y OF L O O P G R O U P S 8 7

Thus each g~G~ has an expression

g=gKgI~gU, gK C~, gaeHk,+, gueJu �9

We wish to show that gK, gH, gu are uniquely determined by g. We fix a basis of V~ which is both or thonormal and coherently ordered (we can do this since the weight spaces are mutual ly orthogonal, with respect to { , }). Then, ~ n H k , + J t r consists of unitary matrices which are upper triangular. I t follows that these matrices must be diagonal. By Corollary ( i4 .18) , H k is the set of diagonal elements in J . Hence Ilk, + is the set of diagonal elements in H k , + J v (since HkC~Jv={I} , by (I4. I6)). Thus

Rt~Hk,+ J u C Hk,+.

But the only unitary element of Hk, + is the identity, and hence

R:nH ,+J ={1}. Since H k , + t ~ J u C H k n J g is the identity, as we just observed, we have:

Lemma (x6 . I4 ) . - - The group GX k decomposes as

with uniqueness of expression.

17. A fundamenta l est imate.

From now on, with the exception of the appendices, we take our field k to be R or C. �9 g ~ ( A ) , Recall from w 6 (following (6 7)) that we introduced the extended k-algebra " ~

and the extended subalgebra e ~ Dk(A). Indeed g~(A)=k| I)~(.~)----kQh~(A), e A where g z ( ) = f l z ( A ) | D~(A)=I)z(A) | are semi-direct products (I)~(A) is

a direct product) . We should remark that the degree derivation D---- Dr+ 1 leaves flz(A) invariant, so that g~(A) is well defined. Also we let D denote the induced derivation on flz(A), and on gk(A). We fix a normal element XED (recall from w 15, that X being normal means that X(h~)4:o for some i = i , . . . , g + i ) , we consider the corresponding highest weight module V~, and the Chevalley group ^x x G k C Aut V~. Recall from Theorem (14. io), that we have the Tits system ^ x (Gk, J , N, S) with Weyl group *gr (more precisely, we saw that N / ( J n N ) was isomorphic to W, which was defined in w 13, after (13�9 i9) , and we identified N/ ( Jc~N) with W). Also recall f rom Pro- position (14.8) that the set {wai(i)}i= 1 ..... t+l C N is a set of coset representatives for the set of reflections S C W, where S is in fact the set of reflections

{r~l,0, . . . , r~t,o, r-~o,1}

(see w 13, preceeding Theorem (13.2o)). x Since k = R or 13, we have from Proposition (6. I I ) , that the representation ~k

is a faithful representation of gk(A) into End Vk x. We let I?~(.~)* denote the k-dual

26


of D~(A). In (3.3) we defined the Weyl group W C A u t e ~ . bk(A) , generated by reflections q, . . . , r t+l, and in w 6, (6 . I4) , we introduced the contragredient action on Dk(A). Identifying 9k(A) with its image under r:,, we have from L e m m a (A. i) in Appendix I of this paper, and from (8. I) that

( I T . X ) A d heI) (X), i=I , . . . , t + i .

Moreover, we have from L e m m a ( i i . 2 ) that for aeAw(A ) and sek*, the operator ha(s ) eAut V~ is represented by a diagonal matrix with respect to a coherently ordered basis. The same is true for heb~(A ). Thus H k centralizes I)~(A) (see L e m m a (I4.2o)) . But H k = N n J (see Proposition (i4. i9) ). Thus ( i 7 . I ) allows us to define a surjection

% : W ~ W ,

where ~o(Wai(I)(dr(3N))=ri, i = I , . . . , l + I .

We examine the action of W on t)k(A ) further. We note that W leaves Ikk(A) invariant, and that

w. D ---- D mod I)k(A), wEW.

Thus for w~W, h~I)k(A), we may define w*h~Dk(A) by

w ( D + h ) = D + w , h .

We also have the surjective Lie algebra homomorphism (see Notational Remark (4.7)) : fl(A) --> 9, given in Theorem (3- 7)- This homomorphism induces a surjective homo-

morphism (which we still denote by ~)

:

where ~(hi) = I~ . , i = i , . . . , g

(ht+l) 0

(see Theorem (3.7)). We now compute "~(w,h) for w~W, heI)k(A). It suffices to compute "~(rl,h )

for i = I , . . . , g-t-i, and in this case we have from a direct computa t ion

( i 7 . 2 ) ~(r i ,h)=~(h)- -e i ( '~(h))Hi , i = 1 , . . . , ~,

~(rt + ~ , h)= W~o(~(h) ) + H~o,

where for eeA(A), we let w~ denote the orthogonal reflection with respect to the hyperplane {HeI?k(A ) ] e(H) = o}.

Now we have identified A with t)I~(A) in w 13. has a natural extension to be(A) (still denoted r~,k). a s

(x7.~,') "~(ri,h)=r~i.o(~(h)) , i = I , . . . , t

h) =

Also, each of the reflections r~, k We may then reformulate (I7.2)

264

THE ARITHMETIC THEORY OF LOOP GROUPS 8 9

Thus, by means of * and the projection ~ : t)k(A)-+I)k(A), we may define a

homomorphism

: W ~ W

such that ~(r~)=r~i,o , i = I , . . . , t,

~)(rt + l)=r__%,l.

But then, thanks to ( i4 .7) , we see that w,i(~)(Jc~N), regarded as an element of W,

is r~,0 for i = I , . . . , g , and r %,~ for i = t - t - I , and hence aP 0 and �9 are inverses of each other. Hence, in particular we have

(x7.3) The homomorphism q ) 0 : W - + W is an isomorphism of W onto W.

From the theory of affine Weyl groups (see for example [9], Proposition (I .2)), it follows that for all hst)n(A) we can find w e W so that

~d'~(w,h))>o, i=~, . . . , I ,

I ~_~ ~O(~(W * h)) 2 o ,

or equivalently

(i7.4) a~(w(D+h) )>o , i = I , . . . , g ,

I >at+~(w(D + h))>o.

We now prove some technical lemmas gk(A)-module V~.

concerning the structure of the

Lemma (x7.5) . - - Let ~I)~(A)* be dominant integral; for i = I , . . . , ~ . Then the set of all vet)R(A)* of the form

t

(~) ,~ = ~ . - - E n ~ , n~eZ, n~2o,

and such that ~ is dominant integral, is finite.

i.e., assume ~(Hi)_>o

Proof. - - For v of the form (~) we have:

t t (v, ~)=(~, - Z n ~ ) + ( ~ , ~ ) - (~ , Z n~)<(~ , ~) ,

i = l ~ 1

since ~ and v are both dominant. But v must vary over the dual lattice to the lattice

generated (over Z) by the coroots. Since the above inequality implies that v must

be of bounded norm, and since ( , ) is positive-definite on t)n(A)* , we obtain the

desired finiteness assertion. �9

265

12

9o H O W A R D G A R L A N D

Lemma (x7.6) . - - For each i = 1 , . . . , tAr-I, and each integer n>o, the set of weights of V ~ of the form

/ + 1

X-- Z nja~, j = l

with ni<n , is finite (1).

Proof. - - Let Oi--{al, . . . , ai, . . . , at+l}; i.e., 0 i is the set of simple roots of A(~) with a i omitted. We let miCg(A) denote the subalgebra generated by the ej, h i , f , with j + i , and let Iki denote the linear span of the h~, j # i . We consider the subspace of V ~

nj~O j ~- i

where X '=X--aa i , with ~ some integer such that o < a < n . Then ~ is an mi-submodule o f V x. As we shall show in Lemma (i7.7) , below, V x is a direct sum of finite-dimensional, mi-submodules, and hence is completely reducible. I t follows that ~ is also a direct sum of finite-dimensional mi-submodules. But thanks to Lemma (i 7-5), V must in fact be a direct sum of finitely many, finite-dimensional m~-submodules, and hence V must be finite-dimensional. Lemma (i7.6) now follows (letting a vary between o and n). �9

We now prove the following assertion, which was used in the proof of Lemma (i 7.6):

Lemma (x 7. 7). - - VX is a direct sum of finite-dimensional, irreducible mi,submodules.

Proof. - - The highest weight vector v0~V ~ is a weight vector for mi (where we take the Cartan subalgebra spanned by the h~, j + i ) , and clearly corresponds to a dominant, integral weight of m~. Thanks to Lemma (7.12), v 0 must generate a finite- dimensional m~-submodule W 0 of V x (see in [8] the remark following Proposition (6.2), p. 6I), and W0 is homogeneous with respect to the weight space decomposition of V x. Let k ~ o be an integer. Assume inductively there exists an mi-submodule WqCV x, which is a direct sum of finite-dimensional m~-submodules, is homogeneous with respect to the weight space decomposition of V ~, and contains V~ for every weight of V x, with d p ( ~ ) < q. Since Wq is homogeneous with respect to the weight space decomposition of V x, the subspace Wq has a well defined orthocomplement W~ with respect to the positive-definite, Hermit ian form { , } (see w 9, and [7], w I2), and W~ is homogeneous with respect to the weight space decomposition of V x. Thanks to (16. I), we have mi=m: . , so W~ is mcinvariant. Let ~0 be a weight of V x such that d p ( ~ 0 ) = q + i , and V x intersects W~ nontrivially. Fix a non-zero ele- Iz0

p X _l_ Vr ment v eV~. (~Wq. Then generates a finite-dimensional mi-submodule W' of W~.

(1) See [8], L e m m a (5 .3 ) .


We then repeat this process with Wq@W' in place of Wq, and so on. We eventually construct in this way, an m~-submodule Wq+ 1C V x which is a direct sum of finite- dimensional irreducible submodules, and such that Wq+ 1 ~ V~, for every weight of V x, with d p ( ~ ) < q + I . This completes the induction, and hence the proof of

Lemma (i 7" 7)" �9

Gk, the subgroup (defined in w 16) generated by We now consider H k , + c ^ x all h~i(s), with s > o and i = I , . . . , g + i . I f • then • has an expression

t ' + l x = H hai(si) hi, ni~>o, si>o, nieZ.

~=1

We define l n • by: / + 1

In x = 2~ n i ln(si)h ~.

By (I7.4) , we have that for all r > o ,

(17.8) ai(w(--rD + ln x))<o,

- - r < a t + l ( w ( - - rD + In • o.

For each weW, we choose w'eN so that

(17.9) (i) w' is a product of elements

(ii) O0(w ' ( Jc~N))= w.

there exists weW such that

i - - I , . . . , g ,

w~i(1), i = I , . . . , t + l .

Since V x is a module for the extended Lie algebra g*(A)=g(A) | and since D preserves V~ (we assumed in w 6 that X(D)~Z), we have that Vk x is a g~(A)-module. For any h~[)~(A), we define eh~Aut V~, by stipulating that e n maps

X e h each weight space into itself and that on V~,k, is the scalar operator e ~(h) We note that for • we have

(I 7 xo) e ln~ •

e Since we have defined e ~ for each h~I)k(A), we have, in particular, defined d D for

each r~k. For each w~W, we fix w '~N as in (17.9). Let • rEk. We then

have, by Lemma ( I I .2 ) ,

(X 7 . IX ) W' >:(W') -1 = e w(ln •

w'(e 'D) (w') -1 --= e ~(D), rek.

Now H k is normalized by N (recall from Proposition (14. i9) , that H k = N n J , so H~ is normal in N by the axioms for a Tits system, (13. 2I), and by Theorem (14. io), which asserts that ^x (Gk, ~ , N, S) is a Tits system). Also, H~, + consists of those elements

in H k with positive eigenvalues (as follows from (16.13)). Hence the first equality

of (I 7. i i) implies that

w'tH ~(w'~-l=Hk k k , + l k ] , + "

267


For • rek, ~ t f ( A ) * , with Vt~Ez, we set

(• ~ = • e- ,~(o),

where x~ek * is defined by

• V - - • v c V ~ ,

if ~z is a weight, and • is then defined for all ~ E z. thanks to (17.1o) , we have

N o w given • +,

then w ' E N as in ( I7 .9 ) . appl ied to e-*I}), we have

{I7 . I2 ) (w'• I, i = 1 , . . . t, e--r~(w'•

Lemma ( x 7 . I 3 ) . - Let xeHk, + and r e R with i = I , . . . , g + l . Then for some j = i , . . . , g + i , we have

(•176 1.

We note that if • e Ilk, +, then,

r ~ R with r > o , we choose w~W as in ( i7 .8 ) , and F rom (17.8) and (17.11) (the second equal i ty being

r > o . Assume (• rD]ai< I ] - - ,

P r o o f . - I f (• for some i = I , . . . , t , there i s n o t h i n g t o p r o v e . Thus, assume (• for each i = i , . . . , g . Then, since a ~ ( D ) = o for i = I , . . . , t ,

we have

(*) X ai~-I, i = I , . . . , ~.

Now • has an expression as a p roduc t g + l

• __I2Ilh~(si)"i , &>o , n~>o in Z.

t + l

We have • = I I S~ iag+l(hi), i = 1

and thanks to ( . ) , this equals one, since when acting on I?(A), at+ 1 is equal to an integral l inear combina t ion of a l , . . . , a t. But then

(• rD) a t +1 = e- ' < 1,

(since r > o ) and L e m m a (17.13) now follows. [] F rom Lemmas (17.6) and (17. I3) , we immedia te ly have

Lemma ( x 7 . i 4 ) . - Let • k and let cek with r=~e(c )>o . Assume •215215

with xleHk,0, • and

(• ,D),~< 1, i = I, . . . , e + I.

Then for all A > o , there is a finite subset of weights of V x, such that i f p. is a weight o f V x

which is not in this finite subset, then

[(• ~D)~I > A .

268


We define the ring J C k as follows:

l Z, if k = R

J = l ring of Gaussian f / integers, if k = C.

93

We let s176 a C 5~ denote the subring of all formal Laurent series in t, with coefficients in J. We fix a Chevalley lattice Vz x in V x, and a coherent Hermit ian structure { , }

on V z, as in the w o o f of Theorem (i ~. I) of [7] (also, see w167 6, 9 of the present paper). We may thus assume that we have fixed a highest weight vector v0oeo in V x so that {v0, v0}=i , and so that { , } also satisfies

{ml, m i ) e Z

{ml, m2}eJ, for mi, m~eVa x.

We let ^ ^x ^ x F = Pk C G = G k denote the subgroup

F={V~OIv(V~)=V~}.

The following is the central result of this section:

Lemma ( x 7 . x S ) . - - For geG, cek, with r=~e(c )>o , we can find 70eP such that

(I 7. I6) {ge-~Vy0v0, ge-~DyoVo}<{ge-cD~'p0, ge-~DyVo} ,

for all yeF.

Remark. - - Lemma (17. 15) remains valid if we replace P by the subgroup P0 generated by ,Z~(,(t)), ~ A ( A ) , a ( t ) ~ a . The same proof (given below) applies.

Proof. - - The argument is based on the Iwasawa decomposition

of Lemma (I6. I4). Thus g e G has a unique expression g = g i g i g v , with g i e ~ , g ~ H k , +, gu~Ju �9

We first note that for any positive number A, we may choose a finite set of weights A = A A of Vk x, such that for ~r a weight of V~, we have

[(gHe--CD)~l>A.

TO see this, we choose weW, and corresponding w'~N (as in (I7.9)) so that q % ( w ' ( J n N ) ) = w and so that

(w'(gv, e-~D)(W')-l)aiSI , i = I , . . . , g,

e-" < (w'(gue-~D) (W')-l)at+' < I

(we have applied ( I7 . I2)) . Now

w'(g~e-,D) (w ' ) - ' = •

269


for some • but those in a certain finite subset, we have

i( w, (gHe- cD) (W')- ~)~[ > A.

The left side of this inequality is just (gHe-~D) ~-~l~l, SO

] (gHe-~ A,

for all but finitely many weights of V ~, since W permutes the weights of V x. have thus found our set A A. Enlarging A A if necessary, we may assume

and thus it follows from Lemma (i 7.14) that for all weights [z of V ~,

are also in A x.

7eF we set mv-~y.v+eVXa. if ~ZeAA, then all weights of V ~, of depth<dp(~z),

For veVk x, we set IlplI~=~,{v,p}, and for y l e F arbitrarily, and let

(x 7. x7) A = Ilge-~Dm~,l[>o.

W e

that

Fix

We set V~, j=J | where V ~ , z = V ~ n V z , for every weight ~ o f V x. For y~F, the vector m v may be written as a sum, with respect to the weight space decomposition

IIv

where ~z runs through the set of weights of V x.

I f m v has a non-zero component mv(~0 ) in Vx~,,J, with ~0r and if we choose ~0 of maximal depth (see w I2) among all ~z such that m v has a non-zero V ~ Iz, J component, then

I]ge-CDmvl [ = [IgRgue-~Dmvl] > I[gHe--~D mv(~0 ) [I

= [(gae-~D)~. I [] mv(~zo)1 [ ~l(gHe-~D)~. [,

since f[mv(ot0)[[ is the (positive) square root of a non-zero, positive integer, and is hence at least one. Summarizing:

I[ge- cDm v ][ >_[(glie- ~D)~~

~z0r A we have

[]ge-~

Now

But since

(xT.xa)

Recall that if ~xeA,, then all weights of V x, of depth <dp([z) are also in At . consider the set E r of those -reF such that mv=y.v+ satisfies

X t'~y E tz ~I~AA Vy. - - df VA �9

Then, thanks to ( I7 . I7) and ( I7 . I8) , we have y I~Er . Also, ( i7 . i8 ) shows that if yCEr, then [[ge-cDmv[]>A. I f we set ~(y)=[[ge-cDmv[[, y~F, then for y e E r ,

we have

~(u = I I gitgu e- cDmv II,

270

THE A R I T H M E T I C T H E O R Y OF LO O P GROUPS 95

where gEgue-CDmveVA. Since V~t is finite-dimensional, qo(7 ) achieves a min imum as 7 varies over Er , say for 7 = 7 0 . Then, since 7 l e E r , we have

~(70)<A,

and thus for 7 6 E r

~(70) < A < l l g e - cDm, l I,

by ( i7 . i8 ) . Thus, for all 7eF , we have

[lge-~Dmv~

This is obviously equivalent to (I 7. I6), and so we have proved Lemma (17.15). �9

I8~ A fundamenta l domain for F n ~u in ~v.

We recall that beginning in the last section we have adopted the notational convention that in the remainder of this paper, with the exception of the appendices, we take our field k to be R or C.

In w 8 we defined the group homomorphism

(I)': G~(A)~Gaa, .v(A ).

Thanks to L e m m a (8. i4) , and the fact that g~(,~) is perfect since c h a r k = o , the homomorphism (I)' is injective. We now note that:

Lemma ( I 8 . I ) . - The homomorphism ^ X

q) 'oAd : Gk ~ G~x,a,(A),

is injective, when restricted to the subgroup Ju C Gs

Pro@ - - I f g e J v , and if O ' o A d ( g ) = e , the identi ty of Gaa,so(A), then g~kernel(Ad), since qS' is injective. But then geAut(Vk x) is a scalar multiple of the identity, by Schur's lemma (Lemma (9 . i ) ) . However, relative to a coherently ordered basis, the element g is represented by an upper triangular matrix with ones on the diagonal (by (I 4. I5) ). Hence g must be the identity. �9

We fix a Chevalley basis of g(A), and order this basis so that positive root vectors are represented in the adjoint representation, by upper triangular matrices. For any commutat ive ring R with unit, we may also regard this Chevalley basis as a basis of gR(A), and of course it is then still true that positive root vectors (in gR(A)) are represented in the adjoint representation, by upper tr iangular matrices.

We then have:

Lemma (I8.2) - - The subgroup O'oAd(Jt~ ) of G~a,~(A ) is the subgroup of all geGad,.~(A) such that relative to the Chevalley basis of g(A), ordered as above, g is represented by a matrix with coeffcients in O, and such that the reduction of g mod t is an upper triangular matrix with ones on the diagonal.

271


Proof. - - From the definition of J~, at the beginning of w I2, we see that (I)'oAd(Ju) is contained in the subgroup J~ of all geG, a,~(A), such that g is represented, relative to our fixed Chevalley basis of g(A), by a matrix with coefficients in (9, and with a reduction mod t which is upper triangular with ones on the diagonal. To prove that JuC(I) 'oAd(Ju) , we first note that, thanks to Theorems (2.5) and (2.24) in Iwahori-Matsumoto [9] (where one interchanges the roles of A+(A) and A_(A)), it suffices to show J~n(@'oAd(T~X))ffO'oAd(Ju) . This is in fact proved in the process of proving (18. i4) , below. �9

From now on we identify Ju with its image J~ C Gaa,.~(A), where this identification is made possible by Lemma (18.I) . For each j > o in Z, we define the subgroup Ju (j) C ~r by

A(u~)----{gcJu[g ~ I rood t~}.

In particular, Ju (~ Ju- We let

F(~)= F n J u (j), j 2 I , j e Z .

We also set F U = F n Ju . We shall show that, for each j_> i in Z, there is an isomorphism of abelian groups

( I8 .3) IF(J) : Jv(J)/Ju (i+~) --% gk(A), ( j > i),

where gk(A) is a group with respect to the vector space addition, such that if we let

~:(5): ju(~)~j(J)/ju(~+l), j > o ,

denote the natural projection, then, for rEk, 0~eA(A), j ~ x , we have

(x8.4) ~F(i) o r:(2 (z~(rti)) = rE~,

W (j) o r: (2 (h~ (I ~- rt ~)) = rH~,

and hence

(,8.5) W(J)o~(~)(F(~ )) D gj(A), j • i .

To prove the existence of the isomorphism ~F (j) satisfying (I8.4) , we consider the ring 0 ~ / t ~ + 1 0 , which, as a vector space over k, has a direct sum decomposition

oj= II t'k. 0 < , < j

Then the algebra goi(A ) has a corresponding direct sum decomposition (as a vector space over k):

(x8.6) g~j(A)= H t~gk(A). 0 < i < j

We fix an ordered basis of g~i(A) relative to this decomposition, so that the basis vectors of goj(A) consist of a union of basis vectors of the subspaces t~gk(A), so that the basis vectors in each tigk(A) appear consecutively, and so that for i<i ' , the basis

272


vectors in tVgk(A) appear after those in tigk(A). Then for g e J v (j), A d g determines a k-linear transformation of gos(A), and relative to our basis, and the direct sum decomposition (i 8.6), the matrix of this linear transformation has a block decomposition

/i ~ 1 i ~ ~ ~ , �9

where Aj(g)eHomk(flk(A), tJgk(A))~Endkflk(A). One sees directly that

( I8 .7 ) Aj(x~(rtJ))---- r ad E~,

Aj(h~(I + rtJ)) = r ad Ha,

where rek, 0~EA(A), j_>i , and thus, identifying gk(A) with its image under the adjoint representation, we see that we may define tF(~) by

~F(J) o rcJ = A i,

and then note that qp(i) satisfies (I8.4) , thanks to (18.7). We also wish to consider the projection 7:~ J u ->Ju / J~ a). With a slight abuse

of notation, we let z~(r), c~eA(A), rek, denote the automorphism of gk(A) defined by

x~(r) = X (ad X~)irj" j>0 j !

We let Ga0,~CGa0,. v denote the subgroup of all elements which, relative to our fixed Chevalley basis of 9(A), are represented by matrices with coefficients in (~. Then 7: ~ is the restriction of the projection (again denoted by ~0) from Gad,, to Ga0,r (1), where G~I,r (1) may be identified with the subgroup GkCAut ilk(A), generated by all z~(r), ~ A ( A ) , rek. Moreover (identifying Jo with a subgroup of GkX), our notation is consistent, in the sense that if ~ A + ( A ) , r~k, then

7:~ = z~(r).

We let UkC Gk denote the subgroup generated by all x~(r), :r rEk.

Then 7:0 (Ju) = u k ,

~~ D the subgroup d U of Uk, generated by all x~(r), 0~A+(A), r~J.

We define a fundamental domain ~ C k, for the action of J (acting as trans- lations) as follows:

~__/x~R, ]x]<__I/2, if

- - t z = x + ~ v , x, y e R ,

k ~ a ,

Ixl<l/ , tyl<i/ , if k = C .

13

273


We define g~(A)Cgk(A) to be the subset of all xegk(A ) such that each coordinate of x, relative to our fixed Chevalley basis is in 9 .

We now fix an order on A+(A). Then every element of U, has a unique expression as a product l'I Z~(s,), s~k, where the product is taken with respect

G A+(A)

to our fixed order on A+(A). We let U ~ C U k consist of all products ~eaY[§

s~e~ (the product again being taken with respect to our fixed order on A+(A)). A straightforward induction then shows:

(I8.8) For all ~ U k , there exists yGdu, such that ~/yeU~.

Now we have fixed an order on A+(A). This order, in turn, determines an order on A_(A) : - -A+(A) . We let

U~ =subgroup of G~ generated by the elements Z~(~(t)),

aeA+(A), a(t) e0,

U~ =subgroup of G~ generated by the elements Z~(a(t)),

:r a(t) e~ .

We also recall, from Lemma (:3.7) , that T~CG~ is the subgroup generated by the elements h~(a(t)), ~eA(A), ~( t )e~, and by the elements h,t+1(s), sek*. We note that in the notation of w 13, we have

(x8.9) Ur = U+,c,

U~ = U_,c.

It thus follows from (13.22) that

(xS. to) J=U~T~zU~,

and, from (I3.23) , that every element of Uo (resp. of U~) has a unique expression as a product

II G A+(A)

(resp. ~ Ha(A) Z~(a~(t)) , a:(t) e~) ,

where the product is taken with respect to our fixed order on A+(A) (resp. on A_(A)). It then follows from (18. io) and the above remarks, that every element x of J has an expression

with ~(t) e0, aeA+(A), ~'~(t) e # , ~eA_( i ) , heT~,

where products are taken with respect to our fixed orders on A+(A). Moreover, we have

274


Lemma ( i8 . i2 ) . - The subgroup . J u C J consists of all x ~ Y such that in an expression of the form (I 8. I I) for x, the element h may be written as a product

g

08. 3) h =

with , i ( t ) - I m o d t ( , i ( t ) E O * ) , i = I , . . . , L For j > i , the subgroup j~Jl consists of all x ~ J u such that in an expression of the form (18.11) for x, we have , ~ ( t ) - o m o d t j, ~ A + ( A ) , and , '~(t)--o mod t j, 0~EA (A), and in the expression (18.13) for h we may assume a~(t) - i m o d t ~, for i = I, . . ., ~.

Proof. - - Recall, we have identified J~l with its image under 49' o Ad. Relative to a Chevalley basis of fl(A), ordered so that positive root vectors are represented by (strictly) upper tr iangular matrices, the elements of Ju acting on ~.~k(A), by means of 49'oAd, are represented mod t by unipotent matrices. Hence, the elements of x T , t ~ J U act on fl~k(A) as the identi ty rood t. On the other hand, for any h~Tr x, we have

g

h = ,=H=h='(=~(t)) rood(kernel O 'oAd) , ,"](t)~r i = I , . . . , t.

Each ~(t) has an expression

~( t )=ai ( t ) s i , siek*, ,i(t)eO*, , ~ ( t ) - I m o d t .

Then we have

h -~ h'h" rood(kernel 49' oAd), g

where h' -~ ,~1 h~i(e'(t))'

r

h" =

Now, if heT~n~'t~ , then

h " e T ~ n Ju mod(kernel 49' o Ad),

since h'~T~c~oCu, by L e m m a (12.2). But then, h" acts on 9.~k(A) as the identity rood t. But 49'oAd(h") is equal to its reduction rood t. Hence h"ekernel(49oAd). But 49'oAd is injective when restricted to ocv, by L e m m a (18.1). Since, by L e m m a (I2.2) , h'~T~noctr, we thus have h = h ' ; i.e., we have proved:

(I8.I4) I f heT~Xn~'u, then h has an expression as in (18.13) with , ~ ( t ) - I m o d t , i = I , . . . , L

But now if xe~Cu, and if we express x as in (18. i i ) , then h e J u n T ~ , and hence the first assertion of L e m m a (I8.12) follows from (18.14).

We now turn to the second assertion of L e m m a (18.12). I t is clear that if xE#" U has an expression (18. II) , with ,~(t), ,'~(t) ---o mod t ~, 0~EA(A), and with h

275

IO0 H O W A R D G A R L A N D

expressed as in (18.13) , with a i ( t ) - I mod t i, then XeJu (~1. The converse for j > ~

follows easily from (18.7) and the result for j = I . For j = I , we have that if x ~ J v and i fx is expressed as in (18. I i), with h as in (18.13) , with ai(t) - i mod t, i = i , . . . , ~, then

II ~ A+(A)

where q= is the constant term of (~=(t), ~eA+(A). If x s J dl) then r:~ and we

must have q ~ = o for all aeA~.(A); i.e., a ~ ( t ) - o m o d t , for =eA+(A). This concludes the proof of Lemma (18. I~). �9

Definition (x8. i5) . - - We let 0~ denote the set of all a(t)= ~ qjt ~ in 0 such

that qje~, and ~*e the set of all o ( t ) = E qjt j in O such that q 0 = I and qje~, j > I . We j~_o

let ~r denote the set of all x e J u such that in the expression ( i S . i i ) for x, with h as in ( I 8 . I 3 ) w i t h e~(t)e~)*, ~(t)---~ moa t , i = i , . . . , [ , w e have as(t), a'_~(t)eOe, for all ~ek+(A) , and a,(t)e~*~, for i = i , . . . , t .

Lemma (x8.x6). - - For all XeJu, there exists -(eF U such that

xyeJu ,~ .

Proof. - - We have observed that ~~ contains d u C Uk, and, thanks to (18.8), we can find yoeP u so that if we express xy0 as in ( i 8 . i i ) , then the constant term of a~(t), eeA+(A), is in ~ . For an integer j > o , we let P(j) denote the following assertion:

P(j): For each integer k with o<k<__j, there is an element y~eF U such that

(i) Yk=Y~+l mod ~uP(k+t),

(ii) There exist elements ~( t ) , e ;~( t ) (eeA+(A)) in 0~, of order j (i.e., whose coefficients of t m are zero for re>j), and with o '~(t) e ~ , and there exist elements ai(t) ( i = I , . . . , t ) in 0~ of order j , so that if xj denotes the corresponding product given by (18. i i), with h given by (I8.13) , then

xyk= xj mod Ju (k+l) , o < k < ] .

We note that our initial comments in the proof serve to verify P(o), with Y0 chosen as above. Assume then for some j > o we have proved P(j) . Then

xyj= x~y~, y~e Jcu i +11.

Moreover, thanks to (18.4) and (18.5) , we can find y ' eP~ +11, such that

~(~ + 11 o rd i + a)(yjy') e g~ (A).

We set Yi+~----u and note that with this choice of y~+~, we have that (i) of P ( j + I )

is satisfied. We write ~F~+l/orc(J+l/(yjy')eg~(A) as t

q~E,+ ~=~qiH~, q~, qie~. a G A ( A )

276

THE ARITHMETIC THEORY OF LOOP GROUPS I O I

Now in (ii) of P(j) we have specified certain e~(t), a;~(t) , ai(t). For P ( j + I ) we replace these elements by

~ ( t ) + q ~ t j+l,

~'_ ~(t) + q_~t j+l,

~ ( t )+q i t j+l,

respectively.

~eA+(A) o~eA+(A)

i = I , . . . , ~ ,

We use these elements to define a corresponding x~+l, by (18.11) and ( I 8 . 1 3 ) , and note that Yk, o < k < ] - ~ - I , and xj+~ satisfy (ii) of P ( j + I ) . We see that moreover, the sequences yjeI'u, xjeJc, e, have well defined t-adic limits yeFu, x e e J u , ~ , respectively, and

X'~ ~ X~.

This proves L e m m a (I8. I6). �9

Remark ( I 8 . I 7 ) . - - Our proof of Lemma ( I 8 . 1 6 ) shows more: I f P0.1jC r u is the subgroup of F~j generated by all X~(*(t)), eeA(A), , ( t ) e J [ [ t ] ] , the ring of power series in t, with coefficients in J, then for all x e J U we may find yeP0,u, so that xu J'u. ~.

x 9. A fundamental domain from Siegel sets.

In accordance with the notational convention adopted in w 17, we continue to take k = R or C. We define ~0>o by

/2 /~/~ , k = R

k = C .

Definition (I9.X). - - For ~ > o we let Ho consist of all he -~D, where heHk,+, r = ~ e c>o and

(he-rD)ai<,,

se t .

i = I , . . . , ~ + I .

Definition ( i 9 . 2 ) . - - For ~>o , we let ~o:g2Ho&'u,~2, and we call ~o a Siegel

We set F0 equal to the subgroup of P generated by all Z~(*(t)), eeA(A), a(t) eoo~~ We have

Theorem (x9.3). - - Let ?̀ be a dominant integral, linear functional in I?~(A)*, and assume k ( h i ) = I , for i = I , . . . , g + 1 . Then for geGXk and cek with r = ~ e c > o , we

can find ~ e P 0 such that

ge- ~Dye ~o..

Remarks. - - (i) Later on we shall see that if ce l l , then we can drop the restriction that ?,(hi)=I for all i = I , . . . , g + I (see Theorem (20.14)) . (ii) Of course

277

to,:, H O W A R D G A R L A N D

Theorem (i9.3) also holds for F in place of F0. We do not know the exact relationship between F and F0. (iii) Needless to say, Theorem (i9.3) is an analogue of Theorem (1.6) in [I]. Indeed our proof is modeled after that in [1], though the present case presents some new technical difficulties (e.g., we must introduce the operator D, and the space V x is infinite-dimensional). (iv) The appearence of D seems to us to be natural. Indeed, the operator D is naturally related to the Fourier expansion of automorphic forms on S12 (R) (see e.g. [6]). The picture which emerges in Theorem (I9.3) , when k = C , may be described as follows: Let ~* denote the Poincar6 upper half plane. For each z ~ * , let f'~Z=eizDf'e--izDcG~, and let

be the isomorphism defined by

%(Y) = e~ZVY e- ~n.

We let F act on C, cX• by

(g, z). y = (g%(y), z).

We let ~ o = a l l ( g , z)eGoZ• * such that gd~D~o. Then Theorem (I9.3) tells us that

(v) When h = C , our methods in fact prove Theorem (i9.3) , whenever J is the ring of integers in a Euclidean, imaginary quadratic field. Of course, for each such J, one must make an appropriate choice for %. Also, one must utilize the following for such a J:

( . ) There exists ~, with o < r such that for all ~ec, there exists ~eJ,

such that 1~--'~1<r

The proof of ( , ) follows from Hardy and Wright, An Introduction to the Theory of Numbers, p. 213, Thin. 246 (and accompanying remarks).

Proof of Theorem (19.3). - - The proof rests upon the existence of minima (as in Lemma ( 1 7 . 1 5 ) ) . We recall that by the remark following the statement of Lemma (I 7.I5) , the lemma also holds for Fo" Thus, given ^x gEGk, we may choose

y0eFo so that

[[ge-~ < [lge-~

yEF o. Now ge-*DyoeCDeG~, and relative to our Iwasawa decomposition of

(x9.4) for all Lemma (i6.14) , we may write this element as

! r I ge- eD'~o e *v = gKgHgu,

where gK eK, gHeIlk,+, and gu ~JU"

choose y1EP0, so that g~2-----d~guY1 u,~ the proof of Lemma ( I8 . I6) ) . We have

Ilge-*D oWlVo I I = Ilge-*DVovoll,

We note that g'u'=ecVg'ue-cVeJu . We

(we are using Remark (i 8. x 7), following

278

T H E A R I T H M E T I C T H E O R Y O F L O O P G R O U P S t o 3

since 71Vo=V o. Hence (I9.4) implies

( I9 .5 ) [[ge-CDgoYlVo[[~[[ge-CDyoy1yVo[[, for all y~Fo.

On the other hand, we have p v t - - e D t v - - e D - ge--*D yo yx = gKgHgu e Y1 = gKgH e g~"

We set this last element equal to g'. We have from (19.5):

( , 9 . 6 ) IIg'VoirSJlg'w0]] ', for all V~F0.

We will have proved Theorem (I9.3) , if we show that g'e~o,. To prove this, we take y=Wa/(I) , i : I , . . . , / + I , in (19.6). The left side of (19.6) is, in any case, equal to

(I9" 7) [(ghe- ~D)X ]2.

On the other hand, thanks to L e m m a (I1.2) and to our assumption that ~,(h~) ---- x, for all i = I , . . . , l + l , we have

(19.8) w~(I).VoeV~_~. , i = 1 , . . . , t ~ - I .

Also, identifying gk(A) with ~kX(gk(.~)) (recall from Proposition (6.1i) , that ~k x is faithful, for k = R or C), we have from (7.8) that

w~(1)-t~aiw~(i)----• for all i = I , . . . , t + I .

We note that v~----wai(I).voeV~_~, by (19.8), and

fl~.VX~CV x~+~, aeA(.~), ~ = w e i g h t of V x,

as one checks directly. Hence

~2--ai . V 0 = W a i ( I ) - - ' ~ . W a / ( I ) . V0

- - = o ,

, X since ~..vi~Vx+~. , and VxX+~=o because X+ai is not a weight of V x (see (6.7)). But then, since w~(i).vo~VxX ~, we have

w~(1) .Vo= (~ + ~ ) ( ~ - ~_.,). Vo = (1 -~_~-~_~) .Vo =--~_~.v0,

and hence

(z9.9) since X(h~) = I

~o~w~(I)Vo=--h~.Vo=--Vo, for i = I , . . . , t + I .

We can now compute the right side of (19.6) for

from (I9.9):

( x 9 . x o )

i ---~ I, . . . , / + I ,

Indeed, we get

' ]]gae g . , . v i lJ2=] (gH e + 1012l (g~e-"D)X] ~ , [Ig w~(x).v0[] 2--- , -,D , , --~D)X--~I2

279

~o4 H O W A R D G A R L A N D

where

p = i the constant term of e~(t), in the expansion (I8.II) for ge , when i = I , . . . , t .

the coefficient of t in ~2,.(t) (~0=highest root of A(A), relative to e l , . . - , e t ) in the expansion ( i 8 . i i ) for g~, when i = t - k i .

Since g~eJu ,~ , we have p e ~ , so

iPl <l,14, k=R k=C,

and if we set p ~ = I / 4 when k----R and I/2 when k = C , we obtain from the fact that (I9.7) is at most equal to the last expression of (19.xo) (by (x9.6)) that

- - Po) <_[ ( g ~ e - r I 2 ,

or (g~le- ri))ai< t ~ 3 ' k = R

k = C .

This proves that g ' e~o . , which, as we noted earlier, is sufficient to

Theorem (I9 .3) . �9

prove

2o. The structure o f ar i thmet ic quot ients (prel iminaries) .

As specified in w 17, we continue in this section, to take our field k to be R or C (though this assumption is not always necessary; e.g., for Propositions (2o. I), (2o.2), and Lemma (2o.4)). We let XieI)k(A) , the dual space of I?~(,~), be defined by the conditions

Xi(hj) = 8i~, i , j = i , . . . , g + x .

The following proposition is an immediate Corollary of L e m m a (15.7):

Proposition (2o. x ). - - I f XED is any normal element, then there is a positive integer m,

such that E,,ziCEz, for each i : : i , . . . , t + i .

The next proposition is an immediate Corollary of L e m m a ( i5 .7) and

Theorem (x 5.9):

Proposition (2o .2) . - - Let X, ~ e D be two normal elements. Then there exists an

integer too>O, such that i f m is any positive multiple of too, m = j m o , j > o , there is a well defined

group homomorphism

rc(X, m~) : .o k ,o h ,

280

THE A R I T H M E T I C THEORY OF LOOP GROUPS IO 5

satisfying (and uniquely determined by) the conditions

(20.3) n(X, m~x)(X~(v(t)))::Z~a((~(t)), aeA(A), a(t)efs ~

The next result will be needed presently;

Lemma ( 2 o . 4 ) . - - If, relative to a coherently ordered basis of V~, the element geG~ is represented by a diagonal matrix, then g is in H x (and, of course, conversely).

Proof. - Let g e J w J , w e W and assume that, relative to a coherently ordered basis, g is represented by a diagonal matrix. We first show that w is the identity. Assume this is not so. Recall that by Lemma (15.2), we have

(20.5) c

and that in w I7, we proved that there is an isomorphism (I)0: W ~ W , of W onto W (WCAut(D~(,7i) *) being the Weyl group of ge(~), defined in w 3--(also see w I7) uniquely defined by the conditions

( 2 0 . 6 ) ~Po(Wai(I)(J(3N))==ri, i = I , . . . , f @I.

By Lemma (11.2), (i), and by (20.6), we see that if n e N represents w, then

( 2 0 . 7 ) x x n(V ,k)

for every weight ~z of V x. Since "~,CE), (see (2o.5)) we therefore have that ~o(w)(~,')4:bd for some weight b~' of V x. The point here, is that since we have assumed w different from the identity, and since (1) 0 is an isomorphism, (1)0(w) cannot be the idcntity. But then (D0(w), restricted to Er, is not thc idcntity (see [8], w 2, the discussion preceeding Proposition (2.5)). Choose a weight b~ of V z of minimal depth among all weights b~' of V x such that (D0(w) (Ez')4: bL' (for thc definition of depth scc the beginning of w I2).

Also, write g e J w J as

g=xny , x, y e J .

If wV~,k, v4:o, then, thanks to our minimali ty condition on the depth of ~z, and to Corollary (14. I8), we have

g.v =cn.v (modulo summands of strictly smaller depth),

where cek*. But, by our choice of ~, and by (~o.7), we have that v and n.v are in different weight spaces. Hence the element g is not represented by a diagonal matrix, relative to a coherently ordered basis; i.e., our assumption that w4:e has led to a contradiction. Hence g e J . But then g e H x, by Corollary (14. i8).

(Gk, J , N, $) is a Tits system. Hence, we Now Theorem ( I4 . Io ) implies that ^x have the Bruhat decomposit ion

tJ Jw , wEW

281 14

xo6 H O W A R D G A R L A N D

(see [3], Chapter IV, w 2, Theorem i, p. 25) , and every element g of G~ is in J w J , for some weW. We thus obtain the Lemma. �9

Now let XeD be a normal element, and let G~'~ C Aut V~ denote the subgroup generated by the elements of G~ and by the elements d D, re R (recall from w 6 that D acts on Vz x (we assumed X(D)EZ) and hence on V~ for every field k, and that d D was defined in w I7). We then have:

Lemma ( 2 0 . 8 ) . - - Let X, [zED be two normal elements and assume 7~ CEx. Then the homomorphism

of Theorem (i 5. 9) has a unique extension to a homomorphism

such that n(X,~)(e m)=e a) for all r~R.

Proof. - - For each r~R, the automorphism e m of V~ normalizes 0~, (v--- or X) and one has

re(X, [x)(e'Dge-rD)=dDrc(X, [z) (g)e -~D, gEG x.

Hence, in order to prove the Lemma, it suffices to show that

(2o.9) Gx n{e'D},E n = {identity}.

Indeed, by L e m m a (2o.4), this intersection is contained in H~. However, (dD)"i-~ 1, for i = I , . . . , t. Then, if e'DsHk x, we must have, as a consequence of these equalities, that (e'D)~t +x=l . But

(erD)at +1 = e',

and thus r = o , and this proves (2o.9) , and hence proves L e m m a (2o.8). �9

For a ~ o , we write ~x (resp. Ho x) for ~o (resp. for Ho C Hk x) whenever we wish to keep track of the X-dependence (see Definitions (19. I) and ( i9 .2) for the definition of ~o and Ho). We let

X, I~ R Ho = Ho

consist of all h'EHo such that

h ' = h e -'D, h~Hk,+, r>o .

We set ^ R

We write F0 ~ for T 0 in ^x Gk, and similarly, , ~ for ,fu, I-I~,+ for Hk.+, and KX for K, whenever we wish to keep track of the X-dependence of these groups. We have

282

Lemma

~ ( x , ~) : "-'k "-'~

(2o. xx)

T H E A R I T H M E T I C T H E O R Y OF L O O P G R O U P S io 7

( 2 o . 1 o ) . - - Let k, ~teD be normal elements such that E~C.~,z. Let We then have be the homomorphism given by Lemma (20.8).

=

Proof. - - From Theorem (I 5 . 9) it follows that

(2o.x2) ~(X, ~)(Z~(cr(t)))=Z~(a(t)), ~eA(A), ~ ( t ) e ~ ,

and then, as a special case of this equality, we have

(2o.i2')

(where we write Z~(S) for Z~(s) ( v = k or ~), to keep track of the dependence on the highest weight). The first equality of (2o. II) follows from (2o. I2) and from the definition of F~ (v=X or ~) in w 17 (after L e m m a ( I7 . I5 ) ) .

In order to prove the second equality of (2o. II) we note that if I ~ C ( ~ ~ ( v = k or ~) is the subgroup of IZ ~ generated by the subgroups W~i(K), i = I , . . . , t + i (see w i6 for the definition of W~, aEAw(A), and of K C SL2(k)) , then the proof given in w i6 for Lemma (I6. I4) also shows that

~ V "# V G~=KoHk,+d~r , v = X or bt,

and then, by the uniqueness assertion of Lemma (I6. I4), we have

-+G,, v = k or ~, a~Aw(A), then On the other hand, if we write tF] for tFo : SLy(k) ^~ from the definition of uF~ in w i6, and from (2o. I2'), we have

+Aw(X) ~F~ -- o ~ , ,

and hence, fi'om (2o. I3),

=

i.e., we have proved the second equality of (2o. i i).

F rom (2o. x2') we also have

7~(X, ~)(hX~i(s)):=h~(s), sek ~ i = i , . . . , tq-x ,

where (as usual) we write h~(s) for h~(s), aeAw(A ), sek*, to keep track of the dcpendencc on the highest weight. The third equality of (2o. II) then follows from this, from the second equality of (2o. Ir) , and from (2o. I2). This proves

Lemma (2o. Io). �9 We can now prove the following extension of Theorem (I9.3):

283

xo8 H O W A R D G A R L A N D

Theorem (2o.x4). - - Let XeD be normal. For any geC_,~, and r>o , we can

find y e F 0 such that - - r D R

ge Y ~ ~Oo"

Pro@ - - Fix p~D, such that p(hi)--~x , for i ~ 1 , . . . , t + I . Then, by Theorem (19.3) , we know that Theorem (20.14) is valid for X=p . I f XeD is any normal element, then by L e m m a (15. 7), Proposition (~o.~) and Lemma (20.8), there is a positive integer m so that

. ~ C .~

E,~ C Ex,

and we have well defined homomorphisms

r:(p, mp): Gg' ~ k ,

r:(X, rap) : Gk x'~ ---> 6 r p' ~.

Using Theorem (19.3), and L e m m a Theorem (2o. 14) is true for X=mp.

(20.IO) (with X=p , a = m p ) we see that Presently we shall prove:

(IO. IS) I f X, ~eD are normal, and if E~C Ex, then the kernel of the homomorphism ; - - ,

is contained in l~ZnHXn (center ^x Gk). We note that (20. I5) , L e m m a (20. io) and Theorem (20. I4) for rap, which we

have proved, then imply Theorem (20.14) for X, thanks to the existence of r:(X, rap). ^) . ^~.

We now prove (20.15). We first note that it suffices to consider ~(X, ~) : Gk-->G k. Then, using the Bruhat decomposit ion corresponding to the Tits system of Theorem (14.1o) (for both G~ and ^~ Gk) we see that first r:(X, ~) respects these Bruhat decompositions, and then, as a consequence, that the kernel of re(X, ~) is contained

^X in J C G k .

On the other hand, we have a commutat ive diagram

(20. x6) r o A~',~ Z/CC'oAd Gaa,~(A)

where, of course, O 'oAd on the right and left are two different homomorphisms. But (either) @'oAd is injective on ~r (by L e m m a (18. i)). Since J = H k J U (semi- direct product) by Lemma (14. 12), by (I 4. 13) and (14.16), we therefore have that kernel r:(X, ~t) is contained in I-Ik-=H ~. Also, by Lemmas (8.14) and (9.1), we have: kernel(O'oAd) CcenterG~, v----X or [z (recall that for k = G or 11, the algebra g~(A) is perfect). Hence, by the commutat iv i ty of (2o. I6), we have:

~X kernel ~(X, ~) C center G k .

Thus, to prove (2o. 15) , we need only verify that kernel r:(X, ~)CN: x.

28~

THE ARITHMETIC THEORY OF LOOP GROUPS io 9

But from (I6 . I3) , we have I-t~=H~.+H~, 0 (~=X or IX), where we introduce the superscript v to keep track of the dependence on the highest weight. Also, we clearly have from the equality

~(X, Ix)(hX~,(s))==h2,(s), sek*, i = I , . . . , t + i ,

which we noted earlier, that

(20. x7) x _ + ) = - +

~) (Hk, o) -- Hk, 0-

Also, H~,+ n H~, 0 consists of diagonalizable automorphisms of V~ (v=X or Ix) each of whose eigenvalues is positive and has modulus one (see w I6). Hence

H~, + c~ H~, 0 = {identity}.

Thus, since kernels(X, IX) CII x, and since IikX0CK z, we see from (2o. I7) that if we show the injectivity of r:(X, Ix) restricted to HkX+, we will have kernel ~(X, Ix)CK x. But the fact that ~(X, Ix) is injective on Hk x, + follows easily from the fact that ~.~ spans I)*(A) *. Hence we have proved (2o. I5), and, as we already noted, we obtain

Theorem (2o.14) from (2o.15). �9

2x. The s tuc ture o f a r i t h m e t i c quot i en t s (conclus ion) .

As specified in w i7, we take our field k in this section, to be R or C. We fix 9et)*(.~)" such that p(hi)=I, i = x , . . . , / + 1 . We set

cw=a+( )nw(A w W,

and we let (~w) denote the sum of all the roots in 4) w. Then from [8], Propo-

sition (2.5), page 5 o, we have

(2x. x ) ((I),o) = p--w. P, for weW.

Thus, for weW, we have an expression for p- -w.p of the form [+1

where the k s are non-negative integers. Let w = r q . . . % be an expression of minimal length of w in terms of the

generators r~. Set

(21.2) b j : ? ' i l . . , rij_l(ai/), j : I , . . . , v.

Then from Proposition (2.2), page 49, in [8], we have @~o--={bl, . . . , b,}, and q)~ has exactly v elements. Hence the b~ are mutually distinct.

For each i = I , . . . , l ~ , - I , we let W~CW denote the subgroup generated by

the r~, j+i . We then have:

285

I I O H O W A R D G A R L A N D

Lemma (2x.3). - - Fix an integer i e { i , . . . , t + i } . l + 1

w . p - - p = - - } l k ~ a ~, k~2o, kj~Z,

and ki>o.

I f w e W is not in Wi, then

Proof. - - Let

w = r i l . . . %

be an expression of minimal length of w in terms of the generators r i. Let • be the smallest integer between one and v, such that i~= i (w is not in Wi, so such a • does exist). Then

b• = rq . . . r,~_~ ( a,~)

= r , . . . r,~_l(a,) / + 1

= Z qjaj, q~+o, j = l

and hence

with k~>o. �9

t + l

p - w . p = ~ % ) = b l + . . . + b ,= Zlkja ~,

Lemma (21.4). - - Fix an ie{ I, . . . , g+x }. Then there is a real number c>o , such l + 1

that i f beA+(A), and b = kja~, with hi>o , then

g + l

Proof. - - Let %cA(A) denote the highest root, and express e0 as a linear combination of the simple roots of A(A):

t

j = l t

We let m=j~=im j. Next, let ~ denote the imaginary root

t

$ 1

and recall the description of A+(~) given in (4.3). Thus, if b eA._(A), then b has an expression

t t + l

b = ( j~l qjai) + n( ~ lmjai ) ,

where wc have sct m r + 1 = I, and where

o<5<__mi, j = I , . . . , l

286

T H E A R I T H M E T I C T H E O R Y O F L O O P G R O U P S I l l

and n_>o. / 4 1

b =

and we let

(2x. 6)

where

(2x .7)

Then

On the other hand, in the statement of Lemma (2I-4), we have set

with ki2>o for our fixed i. We set

I c =

2 ( m + I ) '

q t + l = ~ We have

k i = qj + nmj, j - x , . . . , t + I,

m j > I , qj<mj, j = i , . . . , t + i .

ki qi + nmi t+x - - t+ l , by ( 2 1 . 6 )

~tk~ ( j~t qj) -k- n(m + l )

q i + n >--m+n(m+~)' by ( 2 1 . 7 )

q~+n > - ( m + x ) ( n + I )

I ~-I' I if n = o (then qi=ki > I) m

> n i

and in either case, we obtain (21.5). �9

all Lemma ( 2 1 . 8 ) . - - F i x an

weW--Wi, /f we set / + 1

~ - - W ' P : j = 1

then / + 1

(2~ .9) k,>~(.~ k~).

i E { l , . . . , t - t - I }. There exists • such that for

Proof. - - For a root **+1

f = ~=1 n~a~

in A+(.~), we let h t ( f ) (= height o f f ) 1+1

h t ( f ) = 5~lnj.

be defined by

287

I12 H O W A R D G A R L A N D

The set such that FCA+(A) of all roots feA+(A), / + 1

f =j~a=lnjaj, h i = O ,

is a finite subset. We l e t f l , . . . , f q denote the elements of F, and we set

V-- ~ ht(fo).

Recalling (21.2), and that if w - - r i . . . r i= is an expression of minimal length of w in terms of the generators ri, then @~0={bl, . . . , b~}, we have from (21. I) that

p - - w . p = b l + . . . +b , . t + 1

We write b, = ,-~ik~"a" u = i, . . . , v,

and note that kj = , = kj~.

We set T ( F ) = { u s { I , . . . , ,}[b~r

/ + 1

and M = Y~ ~ kj~. u ET(F) j 1

Then, since wCWi, it follows that M > I . As a result: / + 1 "r / + 1 'v

- ' = k, . ) ( Y, . ) - ' k~'(jl 1 j ~ l u ~ l

>_ ( 5,Fk, , )( M +~)-~

>cM(M+~x) -~, by Lemma (21.4) ,

=6(I + [~ /M)- l~c ( I +~)-~.

We may thus take • --1, in (21.9). �9 For each ~>o, we defined H~ in w 2o, to be the set of all h 'eHo such

that h '=he -~D, heHk,+, r>o. Thus, in particular

(h')~i<,, i = I , � 9 t a - ~ - I .

We now prove:

Lemma ( i I . I O ) . - - For all r there exists M > o such that i f r > M and

h'=he--rDEH~, heHk,+, then for some i = i , . . . , g + I , we have (h')ai<r Conversely,

for all M>o, there exists an r suchthat i f h ' = h e --rI)-Ul~eno, heH~,+, and i f (h')ai<r for some i = I , . . . , t + i , then r>-M.

P r o @ - If h'=he -'v, with r > o

( 2 I . I I ) (h')ai=h ai, i = I , . . . , t,

(h')"t +1 = e- "h~t +1.

and heHk,+, then

THE ARITHMETIC TIIEORY OF LOOP GROUPS z x 3

Writing the highest root %~A(A) as an integral linear combination of simple roots of A(A), we have

l

% = X ms~ j. j = l

For heH~,+, we then have

t ( 2 1 . X 2 ) hal § I Ih -mja j .

J = l

l We let m = 2~ mj. Assume we are given ~>o, and let

Et = ~--m.

Then choose M;>o so that

E'g--M~.

We then consider h ' = h e --'D in H~, with hEHk.+, and r > M . We claim that

(h')"i<e for some i = I , . . . , e + i . I f this is not true for some i = I , . . . , t , then F i > e , for i = : I , . . . , t (by ( 2 i . i i ) ) . Hence, by (~I . i2 ) , we have

hat+l ~ ~'.

But then (h')at+i=(he-'D)~l §

= hat+te-r<r -"

< d e - m < r

Conversely, assume, for some M > o ,

and o < r < M . Then

h ' ~ < , , i = I , . . . , t - - I - I ,

and by (2 i . i 1 ) and (21.I2), we have

hal + t > ~ ',

where ~ ' = , - m . On the other hand

h~t .xe- '=(h ' )~ t . l<~,

by (21 . I I ) and our assumption that h' is in H~. Thus

~' <hal+t< ,e ~.

For i - - - - I , . . . , t , we let m ( i ) = m - - m i and

h"i"i----h-~t+lh -blil, by (2 i . i 2 )

> (~e ~)--1 ~-,,(~/, i = i , . . . , g,

that h ' = h e - 'D is in H~, where hzHk. +

l

b( i )=( Y, mjaj)--mia i. We then have j - 1

289 15

II 4 H O W A R D G A R L A N D

and thus the (h') " i= h ~;, i = I, . . . , g (see ( 9 I . I I ) ) ,

(h')"t+~=h~t+le -" (by (2 I . I I ) )

0" e -'~1,

are bounded below. But

so all the (h')% i = I, . . . , / + I , are bounded below, and this proves Lemma (21. IO). �9

In the proof of Lemma (2o.8), we showed that if we write gEG x'~ as

(~'X. I3) g g, e,D, , ^ x : g eGk, reR,

then g' and r are uniquely determined by g (see (2o.9)). It follows from this fact, and from the Iwasawa decomposition of Lemma (i6. i4) , that in the decomposition

�9 6o = KHo Jo.~, 0>0, (2z ' 4 ) " ^ "

we have uniqueness of expression. For ~"" xawo, we let

X=XKXHXu, XK~K. , XllUI"Ia R, XUE,~,~U.,~ ,

denote the decomposition of x, relative to the decomposition (2I. 14). The operator e rD, r~R, normalizes ^x Gk, and hence, from the uniqueness of the

decomposition (2I. I3) , we have:

^), t ( 2 I . I 5 ) I f gl, g2, gaeGk, if r , r " e R ,

and if gle-r'Dg~-=g3e - ,"D, then r ' = r " .

For each i = x , . . . , / + i , we let W i C W denote the subgroup generated by the elements w , f l x ) ( JnN) , with j + i . For FC{I , . . . , / + x } , we let W v = i p v W i , and we denote by Pr C~x the parabolic subgroup

P~. = J W v J .

Indeed, the fact that PF is a subgroup of ~x follows from Theorem (I 4. io) and from standard properties of Tits systems (see [3], Chapter 4, Theorem 3, page 27).

Theorem (2x.x6). - - Fix 0>0. Then there exists ~>o such that i f x, y e ~ ,

/ f VETo, i f for some i0=x , . . . , t + I ,

(2x. x7) x~l'< ~,

and if

(2, . x8 ) xy = y ,

then y~ Pi,.

Proof. - - We first note that as a consequence of Lemma (2I. IO) and of (2I. I5) , we have that for all ~ '>o, there exists ~>o suchthat if x . y ~ , 7eF 0 satisfy (21.17)

290

T H E A R I T H M E T I C T H E O R Y OF L O O P G R O U P S xx5

and (2I . I8) , then y~{<r for some j = i , . . . , g + I . Thus, in order to prove Theorem (2x.16), it suffices to prove

( ~ I , I g ) There exists ~ o , such that if x, y e ~ , if yeFo, if for some

io,Joe{I, . . . , g + I } , we have

and if x y = y , then yeP{~..~.}.

Now if x, y e ~ , y e F o satisfy (2I. i8), we have

( 2 I . 2 0 ) XKXltXu'~ = Y K Y H Y u "

G k C Aut Vk x, where k e D Moreover, -(.v 0 = m v e V ] (see w 17 and recall we are taking ^x is normal). I f [z is a weight of V x, then the weight component mv(~z ) of m v in V~ is also in V] (see (6.3)) . Thus, if [z 0 is a weight of V x of maximal depth such that mv(a0)4:o, we have

II Vo!l = !l x . x , . m, 11 >_ll !l

On the other hand, from (~1.2o), we have

l] XK XH Xu y " Vo ][ = [ ]YKYnY~" . VO II ---- YXa "

Thus, if t + l

~o=X-- X q~a i, q ~ Z , q~>o,

I t -1

(see (6.7)), and if we set Y= ~ q~a~, we then have i ' l l

yH > X H X tI �9 (2x.2x) x x

Now in w I7, we defined an isomorphism

q~o : W ~ W ,

and hence we may identify W with W, by mcans of r We obscrvc that if y E J w J ,

weW, we may take t + l

~o = X- - 52 q~a i = w. X, i--1

thanks to Lemma ( i i .2), Corollary (I 4. I8) and the definition of r On the othcr hand, interchanging the roles of x and y, wc also obtain

l + l

(ax.ax') XH?>yHyn ~ , x X - where w-~(?,)=X--~, with Y= 52p~a~, p , ~ Z , p ~ > o . i - 1

Combining (2 I. 2 i) and (2 i . 2 I'), we find l + l l + l

a b (~x.22) I<XHYH, "Y= 7~ q~a~, "b= Z pia~, i = l {=1

pi, q~eZ, pi, q~>o, i = l , . . . , ; + I ,

w-'(x)=x-Y. 291


Now assume X is a multiple of p; i.e., assume that for some positive in teger n, we have

n, i = 1, . . . , t +

To prove (21.I9) for this choice of X, we first note that from L e m m a (21.3) , Lemma (21.8) and (21.22), we can conclude that there exists d > o so that if (21.17') holds for a' in place of ~, and if xy = y , then either y is in P~. or y is in P~0. Fixing such an d, we may in particular, assume y lies in one of finitely many double cosets oCwJ, wEW. We then obtain from Lemma (21-3) and from (21.22) that there exists ~>o so that if (21.17') holds, and if x y = y , then y is in both Pro and Pj0; i.e., we obtain (2I. 19), and hence Theorem (21. i6), for X=np. The theorem for general X then follows in a manner parallel to the proof of Theorem (2o. 14). �9

R R rD(-~ ~ X For r>o , we set ~ ( r ) = ~ e G k and P ,=e -~DF0dDC^x Gk. Then

62=

and we also have

Corollary 1. - - For r suffciently large and yeF~

~ ( r ) y n ~ ( r ) , o,

we have y e P i for some i = I , . . . , g + I .

such that

Proof. - - This follows from Theorem (2I. I6), from L e m m a (21. IO) and from the fact that e-rDPidD=P~.

Corollary 2. - - For r suffciently large, f ' , acts on ~ / ~ x with finite isotropy groups.

Proof. - - This follows from Corollary i, and from the fact that F rnP i intersected with a conjugate of K is finite. Indeed, G = K J = K P ~ (see Theorem ( i6 .8)) , and hence our last assertion is equivalent to the assertion that the intersection of K c~Pi with a conjugate p (FrnPi )p -1, pePi , is finite. But if MiC P~ is the finite-dimensional subgroup generated by the Z• j # i , sek, then K n P ~ C M i is compact , and p(F~c~P~)p-lc~Mi is discrete. �9

Corollary 3. - - For r suffciently large, f ' , is not conjugate to Fo by an element of G~.̂ x

Pro@ - - The group F 0 n K is infinite, as it contains all wa(i), aeA(A). Hence Corollary 3 follows from Corollary 2. �9

292

A p p e n d i x �9

We note that D, the g + i s t degree derivation of g(A), maps gz(A) into itself. Hence, as in w 6, we may adjoin D to gz(A), to obtain the extended integral algebra g ~ ( A ) = g ~ = g z ( A ) | We then set I?~,(A)=I)~=Igz(A)| and for a commutat ive ring (with unit) R, we let

I?~(A) = t)~ := R | z I)~,

g~(X) = g~= RQzg~.

For a~Aw(A), seR, we let ~/~(s) denote the automorphism of g~ defined by

~ = Y, s i ad(~/ j ! ) . j2.o

For seR*, the group of units of R, wc sct

wo(s)=~o(,)~o(-s Wo(s),

~o(s) = ~o(s)~o(i) --1

We then have:

Lemma ( A . I ) . - For hetJ~, and seR*, aeAw(A),

~ ) ~ ( s ) ( h ) = h - - a ( h ) h ~ .

Proof. - - The following equalities are obtained from a direct computat ion:

e/o(,) (h )= h-s~(h)~o,

~ - a ( - - s - l ) (h - - s a ( h ) ~a) = h - - a (h ) h a-- sa (h ) E.a,

O~:l;(S) ( h - - a (h )ha - - s(t(h)~a) = h - - a (h)h a .

The lemma follows. �9

We now proceed to prove Lemma ( i i . 2 ) . First we take s to be an indeterminate over Z, and consider

v~, , ~_,~ = z Is, s-*] % v ~ ,

and gz[~y, l (A)--Z[s ,s- t ] | We fix an embedding of Z[s , s -~] in C, and x thus obtain embeddings of Vz[~,~-~j, .qz[.,.~-,j(A) in V x, go(A), respectively. We

x may define w~(s), h~(s)eAut(V~), as automorphisms of V~ which leave Vz[~y, ] invariant. Thus, if we can prove Lemma (I1.2) over C, we may then specialize s and so obtain Lemma (i 1.2) over an arbitrary field k.

293

zi8 H O W A R D G A R L A N D

So now take k = C . Our argument follows that in Steinberg [2I], Lemma 19, page 27. For s~C, wV~, we set v ( s )=w, ( s ) . v , and we note that we may write v(s) as

( A . 2 ) v(s)= Y, sivj (finite sum), vjeV~+j~x . j E z

On the other hand, noting that Lemma for heI)*(A), we have

~X(h). (w~(s) . v) = r~(~)(h)w~(s), v,

where r~(~) ---- ~-- ~(h,)a.

wo(s) (h), (A. I ) implies ~ ( s ) ( h ) = ~ -1

Hence, in (A. 2), we have vj----- o unless q(~)----- [~ +ja; i.e., unless j - = - - ~(h,). Thus, taking v ' = v j , for j = - - ~ ( h a ) , we obtain Lemma (11.2) (i).

To prove L e m m a ( I I .2 ) (ii), we first note that, by a direct computat ion from

the definition of w~(s) (see (7.23)), we have

w~(s) - 1 = w o ( - s ) ,

and thus for, v~V~ x,

(A. 3) h~(s) .v = w ~ ( - - s ) - l w o ( - - i ) . v.

Now, by L e m m a (11.2) (i),

w o ( - ~ ) . v = ( - ~ ) - ~ ( h o ) v '

w o ( - s ) .v = ( _ s ) - ~(h,/v',

and it then follows from (A.3) that

h i s ) . v = s ~lha). v,

so we obtain (ii) of L e m m a (11.2). �9 We now wish to prove an analogue of Lemma (i i . 2) for the adjoint representation:

Lemma (A.4) . - - Let k be any field, and let a, b~Aw(A); then in ilk(A) we have

where sEk, y- - - -y (b , a )=+I is independent of s and k, with y ( b , a ) = y ( b , - - a ) , and

where rb(a ) = a--a(hb)b.

Proof. - - For the most part, the proof of L e m m a (A.4) is parallel to that of Lemma ( i i . 2 ) , so we only give a sketch of the proof. Thus, just as in the proof of Lemma ( i i . 2 ) , we may take k----C and then show that

�9 ~(s) (~o) = v s - " C ~ o ~ ,

for some y~C. But taking s = I , and noting that ~b(I)(~) is in gz(.~), and is in fact a primitive vector, since ~b(I) is an automorphism of 9z(2~), we see that y = ~ i .

294

THE ARITHMETIC THEORY OF LOOP GROUPS x I 9

Finally, noting that [~ , ~_~]=h~, and then applying N~(I) to both sides of this equality, we obtain (using Lemma (A.I)) ;

y(b, a)y(b, -- a)h~(~)= h~-- b (h~)h b =- hr,(a),

and so y(b, a) = y(b, -- a). The independence assertion is clear, and so we obtain

the Lemma. �9

Appendix H

In this appendix, we take k to be the finite field with q elements. We wish to

prove

Theorem (B . I ) . - - Assume x e D is normal, and that X(hi) is divisible by q - - i , for

each i = i, . . . , t -+- 1. Then the homomorphism r: 1 : G} ~ G~k of w ~ 2 is an isomorphism.

Pro@ - - By definition of 7h, we know that the kernel of rc 1 is the group (3 generated by the central elements

h~(a~)h~,(a2)h~,(a~%) -~, eeA(A) , oh, a2e~*.

By Lemma (io. i), we have the homomorphism

- - + G k , ~ , : E(G~k) ^x

defined by

u2"*(Z~(.))=Z~,(.), e tA(A) , ~ e ~ f k.

We let C~CE(G~@ be the subgroup generated by the central elements e e e --1 * ha((~l)ha((rz)ha(cr1(T2) , aeA(A), oh, ~e~LP~.

I f ~ is a fixed long root in A(A), then by Moore [i8], Lemma (8.2), C=~Fe(C e) is generated by the elements

b ~ ( ~ ) ---- h~(~,)h~(~)h~(~l~) -1, r ~ ~ ' .

By Theorem (I2.~4), we have

b[3(15"1, ~ 2 ) = CT(ffl, ~2) ~ I ,

where I denotes the identity operator of Vk x, where ~---27`(h~)(~, ~)-1, and where c~( , ) denotes the tame symbol (see (I2.2o)). However, the only thing we have to note here is that cT(th, ,~)ek*. Now 2h~/(e, ~) is an integral linear combination of hi, . . . , ht+ 1 (see the discussion preceeding Remark (4.7)). Hence, since ?,(hi)

is divisible by q- - i, for each i -= i, . . . , t + I, we must have

cT(~i, ~ ) ~- - I,

and ~i is injective. Since r h is in any case surjective, we obtain the Theorem. �9

29 5

1 2 0 H O W A R D G A R L A N D

The locally compact group G~ k admits a certain irreducible, un i ta ry represen-

tation, called the special representat ion (see [2]), which is an analogue of the

Steinberg representat ion of a finite Chevalley group. On the other hand, when ) ~ D

satisfies the conditions of Theorem (B. I), we have a well-defined modula r representat ion

~ = r~i -~ o r~ 2 : G . ~ ~ G~ C a u t V~,

where ~ : G.~k~G~k is defined in w I2. In analogy with known results concerning

the Steinberg representation of a finite Chevalley group, we propose:

Conjecture.- Let p e D satisfy p(h i )=i , for i = I , . . . , [ + i . The special repre-

sentation of G~ k admits an integral subrepresentat ion such that , when we tensor this integral representat ion with k, we obtain the modula r representat ion zc (q-~/~ (~).

A p p e n d i x H I

In this appendix we wish to discuss how one can extend some of the results of

this paper to the non-split case. The idea is to use the universal properties of our

central extensions in order to lift Galois automorphisms, and then use descent.

I n this section we take k to be a field of characteristic zero. We let 9 denote

a split, simple Lie algebra over k, and we let ~ c = gk denote the infinite-dimensional

k-Lie algebra

We define

: a~ - - ' k

by

( C i ) ~0(~1, as) = residue(aide,) , e l , a , ~ .

Then the % of w 2 is jus t the restriction to k[t, t-1]• t -1] of the v0 defined here.

We then define

�9 : x

exactly as in w 2; i.e., we set

(C2) V(al| , a 2 | (~)(x,y), (~1, a2e's x, ye '~/ .

T h e n - :eZ*(~ c, k) (where we regard k as a trivial ~r module) , and corresponding to v

we have the central extension N

( C 3 ) o -+ k -+ gk gk o,

(1) Recently, J. Annon has proved a modified, topological version of this conjecture. Roughly speaking, Annon introduces a topology on V(q -1)p, and then proves that a k-form of the special representation is isomorphic to a dense subrepresentation of r:(a -a)~ (the density being with respect to Annon's topology).

296

THE A R I T H M E T I C T H E O R Y OF LOOP GROUPS I O I

where we take H e (C4) g~=gkOk (as a vector space over k),

with multiplication defined by

(c5) s), s')] s, s' k;

and where c% is the projection of ~ = gk Ok onto the first factor. We note that ~ is perfect. Indeed, it is obvious that ~'~ is perfect, since g is

perfect. But then it suffices to show the commutator subalgebra of fi~ contains the direct summand k of ((34). For this we need only check that -~(~, ~') is not zero for some pair ~, ~'. But if xeg is any element such that (x, x)+o, then

"~(t| t-~| 4:o,

by (C2). Thus we have proved:

Lemma (116). - - The Lie algebra ~ is perfect.

We may now recall the comments of Remarks (5.1 i) to the present context. Thus we let gr C ~'~ denote the k-subalgebra

g o = 0|

where 0 = 0 k C ~ is the subring of formal power series (see Remarks (5.11)). By a congruence subalgebra of level n in g~ we mean the subalgebra of all ~ g o such that ~ = o m o d t ". We then note

Lemma (C7). - - The central extension (C3) is universal in the category of all central extensions of "ff~ which split over the congruence subalgebra of level n, for some n.

The proof is as in w 2, with the one additional point discussed in Remarks (5-I I). Since ~'~ = ~ | g, we may topologize ~'~ by means of the t-adic topology of . ~ .

We call fi~, the universal topological covering of ~'~.

Lemma (C8). - - For every automorphism

: g ; - + g L

with ~ and ,if-1 continuous, there is a unique automorphism

such that the diagram

~k gk

is commutative.

297

16

I 2 2 H O W A R D G A R L A N D

Proof. - - The existence of ~ follows from the universality property of Lemma (C7). The uniqueness of ~ follows from Lemma (1.5) and Lemma (C6). �9

Now let k ' D k be a Galois extension. Then we have -~k, ~ and ~ , is a Galois extension of ~ . Moreover, we have a natural inclusion

((]9) Gal (k'/k) ~ G a l ( ~ , / ~ )

of the Galois group of k' over k into the Galois group of ~ , over ~fk, and this inclusion is an isomorphism, as one sees immediately from Galois theory. We now let 1 denote an absolutely simple Lie algebra which is defined over ~fk and which splits over ~fk'. We let ~ , denote the Lie algebra

By assumption, we have an isomorphism of Lie algebras over ~Lf k, (and hence over k')

where g is a simple Lie algebra which is defined and split over k. We fix the isomor-

phism in (CI I).

We now wish to show that we can lift Galois automorphisms of g~=~ , | to the corresponding central extension ~, of (C3). Thus, let • be an element of Gal(~fk,/s and let ~:~k~,-->~k ~, be defined by

eI,

(see (Cio)), Then we have

Lemma (CI2) . - - For every xEG',d(~s that ~ preserves brackets and sums, such that

(c,a) s k',

and such that the diagram

l~k' ?k'

*~a e f~a e gk" > gk'

there exists a unique : "~' ~k' such

is commutative.

Proof. - - Assume ~ and ~-1 are continuous (relative to the isomorphism (CII) and to the t-adic topology introduced just before Lemma (C8)). We note that

preserves brackets and that

(cI4) s k',

298

THE A R I T H M E T I C T H E O R Y OF LO O P GROUPS

We wish to define a second such mapping

x ~ : fie, -+ fl~,.

t~3

To do this we use the isomorphism (CII):

(cx5) ~(~|215174 ~ , , x~g.

Then x # preserves brackets and sums, and satisfies

(C16) x~(sv~) ~-~ x(s)• s~k, ~qE'~.

But then ~o(• -1 still preserves brackets and sums, and is k'-linear by (CI4) and by (Ci6); i.e., ~o(• is a Lie algebra endomorphism (over k') of ~ , . In fact, and x ~ are each clearly bijective, so ~o(• -1 is a Lie algebra automorphism of ~k ~, (as a Lie algebra over k'). But • and (• are clearly continuous, and hence, so are (~'o(• • by our assumption that ~ and ~-1 are continuous. Hence, by

~ e . ._> e Lemma (C8), there is an automorphism ~*;gk, ilk' such that the diagram

-~,

A C h e x~ : gk'--~ ilk' be defined, relative to is commutative. On the other hand, if we let the decomposition (C4)

A C - - ~ e t

ilk' - - gk' @ k ,

by

~((~, s))=(~(~), • ~L~,, s~k',

then ~ preserves brackets (as one checks from (C5)) and sums (as one checks directly), and thanks to (CI6), one has

~(s~)=•215 s~k', ~ , .

Moreover, from the definition of ~ , we have

~ , o ~ = • o ~ k ' ,

and hence, if we set

~= ~*o ~,

we obtain the Lemma.

299

,2 4 H O W A R D G A R L A N D

The only remaining point then, is to show that ~ and ~ - t To see this, we consider the tensor product decomposition (CII) basis X 1 , . . . , X~ ( n = d i m fl) of 9- We then have

~(x)--x x; 24 i 1 i ,

where ~j~e~, , i , j = I , . . . , n . But then if

X - -

is a general element of ~ f k , |

Lemma (C I ")))

z 'x where

are continuous. and we fix a

we have from the definition of ~ (preceeding

{~I, ..., tZ.

Since aj and • have the same t-adic absolute value (• only acts on the coefficients of the Laurent series aj), we obtain the desired continuity of ~. The same argument shows that ~ - t is continuous, and hence we obtain the lemma. �9

We let ~----Gal(Cfk,/ocfk)"=Gal(k'/k), and note that if we consider the g-module k',

then we have

(C,7) HI(~, k ' )=o .

This is a simple consequence of the facts that f~ is finite and k' has characteristic zero. Now, consider the central extension

~k ' N c o - ~ k ' ~ ~, - + gk' - ~ o .

A A c ~ C Then, by Lemma (CI2), for every • we have maps • ~ of gk', ilk', respectively, which preserve sums and brackets, satisfy (CI3) , (CI4) , respectively, and satisfy ~k ,o~=~o~k , . In particular, note that for k'C'~,, we have from (CI3):

(CI8) ~($) = • ~(i) .

Since ~ is unique (given • we have

Thus, we obtain the following from (Ci8):

i.e., • is an element of Z~(~r k'*), the one-cocycles of ~ with coefficients in k'*.

By Hilbert's theorem 9 o, there exists soek'*, such that, for all •

( C I g ) ~ ( I ) = • 1.

300

But then

and hence


~(SO1)=• by (CI8)

=•215 by (CI9) = $O 1

~(SSol)=• by (CI3) = • 1,

by the above computation; i.e., we have

(C2o) ~(SSo~)=• 1, s~k'.

We let i denote the k-subalgebra of ~,

(C2z) i ' = { ~ , [ ~ ( ~ ) = ~ , for all

We also have, of course

I={~g~, I~(~)=~, for all

•

•

Hence ~k' induces a homomorphism of k-Lie algebras

~: : i - - > I .

We now wish to show that 7: is surjective, and has kernel isomorphic to k. if ~eI, there exists ~'e~,, such that

~%(~')=~.

But then, since ~ is ~-invariant,

=~k'(~ ), for all ~.

Utilizing the decomposition fi~,----~~ ' flk,| of (C4) , we may take e~, c ~eg,' , and we see that for • there exists s(• such that

~(~') = (4, s(•

But then for • •

•215 ) = (4, s(~-~)sol),

and on the other hand

• • ) = •215 ) = ~1(~, s (• ~) = (4, S(• 1 + ~(s ( • =(4, (s(xl)-b•215 by (C~o).

Thus, •215 satisfies the cocycle identity

s(•215215215215 • •

and hence by (CI7), there exists slek' such that

(C22) s(•215 •162

x25

Indeed,

4 ' = (4, o),

301

t~6 H O W A R D G A R L A N D

But then for

and so

• (~, S~So~) ) = (~, s(~.)so ~ + ~(s~so~) ) = (~, ( s~ - • ~ + ~(s~)so~),

= (~, s~so~),

by (C~o) and (C22)

(~, SlSo 1) ~',

Moreover we have

~((~, S~So~))= ~,((~, S~So~))= ~,

and thus we have shown z: is surjective. Moreover, thanks to (C2o), we see that kernel z~={(o, sso ~) [s~k} is isomorphic to k. Hence we obtain the central extension

( c~3 ) o -~ k -+ ~" -~ I -+ o.

We summarize what we have just proved in:

Theorem (C24). - - Let I denote an absolutely simple Lie algebra which is defined over

and splits over the unramified Galois extension .LPk, , with ~ , | ~ ~ ~ k' -~k " = ~k" We let ICily, ~C ~ C

be the k-subaIgebra defined in (C2I), and we let r: : ' I ~ I denote the restriction of "~k' : gk'-+gk ' '

Then z: is surjective, and kernel r :~k . We thus obtain the central extension (C23) of I.

Remark. - - We can generalize Theorem (C24) to the case when I is semi-simple, and we will now sketch the necessary argument. In this more general case, we have .~fk, | where ~'~ is a direct sum of simple components. We then let fi~, be the corresponding direct sum of universal topological coverings of these

components, and note that, in an obvious sense, fi~,, is the universal topological covering o f ~ # . Then, as in Lemma (CI2), the action of g on ~ , lifts to ~, (we let ~ denote

the lift of • g) , and we again let "I denote the fixed points in fi~,, of g . As before,

the projection ~ , : gk, gk, induces a Lie algebra homomorphism 7: : I-+ I. Since H i ( g , V ) = o whenever V is a finite-dimensional vector space over k with g-act ion

(recall char. k is now assumed equal to zero), we obtain that ~ is surjective. Moreover, if n=dimk,(kernel c0"k,), then dimk(kernel z:)=n. This last assertion follows from

( .) Let V' be a finite-dimensional vector space over k', and assume we are given a g-act ion on V', where each element of g acts as a semi-linear automorphism;

i.e., for a ~ g , we have

~(~v)= ~(~)~(v), ~ k ' , v~V',

where a(~) denotes the Galois action of ~ on 0c. Let V s V' be the space of fixed

points of this action; then V'=~k'|

302

THE A R I T H M E T I C T H E O R Y OF LO O P GROUPS ~27

The following proof of ( , ) was communicated to us by T. Tamagawa: Let c%, . . . , ~r be a basis o fk ' over k, let x~V', and for each i = I , . . . , r, let

y i = Z a G ~

Since (~(~0~))i,o is a non-singular r • matrix, it follows that x is a linear combination of the Yi. Since x was arbitrary, and since the Yi are in V, we obtain (*).

We will continue to use the notation suggested by the last Remark. Thus if g is a semi-simple Lie algebra defined and split over k, and if ~ , = ~ ' | then we let fi~, be the direct sum of the universal topological coverings of the direct summands of ~c gk, gk' denote the projection map. gk,, and we let ~k': ^~ ~ ~ c

Our next topic is to investigate the possibility of obtaining an analogue of Lemma (CI2), and of Theorem (C24), for the groups ^x G k. In view of Theorem (I2.24), one expects such an analogue to be intimately connected with some universal property of the tame symbol. However, here we take a different approach, and continue to utilize the Lie algebra point of view.

We begin with some representation theory. We let g denote a semi-simple Lie algebra which is defined and split over k. We fix a Cartan subalgebra D C g, and let A denote the set of roots of g with respect to D. We fix an order on A and let A+ denote the corresponding set of-4- roots. We let b C g denote the Borel subalgebra spanned by D and the positive root vectors. We set g ' = k ' | , b ' = k ' | and I ) '=k ' | ).

We let iC 0 k,| (resp. iuC 0 k,| denote the k'-subalgebra of elements whose reduction mod t is in b' (resp. is in [b', b']). We then set

(C25) ~' = ~ ; '(~'),

e ~ a

= ( i u ) .

We remark that if g' is simple, with corresponding classical Cartan matrix A, then ~, is just the algebra g~,(A) of w 5. In general, g' decomposes into a direct sum of simple algebras g~, ir each with corresponding classical Cartan matrix A~. We write A for the direct sum matrix of the A~ (so A is the Cartan matrix corresponding to g'), and we let A denote the direct sum matrix of the affine matrices Ai. We then write g~,(A) for the direct sum [I g~,(~.). Then the algebra ~, is isomorphic

iEIo

to g~,(,~), and we let p: g~,(X)-+~, denote the isomorphism. We set I) ' (X)=p-~(~') ,

i = p - a ( i ) , iu=p-~(tv) . We may regard gk,(")CA as the completion of the Kac-Moody algebra (see w 3) gk'(A) �9 Then I)'(A) is contained in gk,(A), and in fact may be

taken to be the Cartan subalgebra defined in w 3. We say that Xe(~')* (the dual

space of ~') is dominant integral, in case Xo peI)'(A)* is dominant integral, and that X is normal if Xope[)'(A)* is normal (see w I5) in the sense that for each i~I, Xop]~,(2i)

303

~28 H O W A R D G A R L A N D

is normal. We let f) denote the set of dominant integral elements of ('~')'. Then

given XEf), we let M(X) denote the one dimensional "[-module, with generator v0, and defined by

v0 = (o,

We let ,~rM(X)__ ^o ^ --k --q/(gk,)| M(X), and note that left multiplication gives --kVM(z) a q/(~,)- (and hence a ~,-) module structure. We let v 0 also denote I| 0 in v~(x) We then let VkX,, be the quotient of--k'~rM(x) by the (unique) maximal ~,-submodule, not intersecting h'v0, and we let v o also denote the image of ~o" ~=~TM(z)--k' in Vk,.x We may regard V~,, as either a g~,- or ~,(,~)- module, and hence also as a gk' (A)-module. The same holds for ~r~(x) �9 k ' ~

Now each Ai, ieI0, is an g i • i matrix for some positive integer [i. We

set l ' = ~ ( l i+ I), so A is an t ' • matrix. We let e~, . . . , ee,,f l , . . . , f t , , hi, . . . , hr

be the canonical generators of the algebra gk,(A), given by the Kac-Moody construction (see w 3). When we regard V~, as a gk,(A)-module, as above, then V~ x, is quasisimple in the sense that there exists a positive integer n such that fi". v0= o for all i---I, . . . , l ' (see [8], w 6). Indeed, since X(h~)2o, and X(hi.)eZ , for each i = I , . . , l ' (i.e., since X is dominant integral), we may take n---- max (X(hi)-k-I).

' i = l . . . . . / '

Also since X is dominant integral, it follows from a theorem of Kac (see [8], Corol- lary (9.8)) that V~ x, is the unique quasisimple gk,(.~)-module with highest weight k. We can then use V~ x, to construct Chevalley groups (~,x, exactly as in w 7 (of course, we now allow that A correspond to any semisimple algebra).

We now return to the context of Theorem (C24), and of L e m m a (CI2). Our goal is, for each

to define a bijection ~ : VkX,-+V~ x, (actually--see (C34) below---one must replace V~ x, by a direct sum of V~','s, in general) such that ~x preserves sums, and satisfies

( c 2 6 ) s k', wV ,, ~(~v)=~(~)~(v), ~eg;,, veV~,.

I f A is a classical Cartan matrix (and so, corresponds to a simple Lie algebra), we have defined, in w Io, the simply connected Chevalley group G~k,(A ). More generally, if A is the finite direct sum of the classical Cartan matrices Ai, ieIo, we let G.~,~,(A) be the direct product

%,(A)=

We now consider ~(~), the image o f t under ~. From the results of Bruhat and

Tits (see [5]) we know that ~ ( ' ~ = A d ~('~, for some ~eG~,~,(A).

304

THE ARITHMETIC THEORY OF LOOP GROUPS x2 9

We let ~x: ~,_+EndVkX, denote the representation corresponding to the

^x c ~ x , ^ ~ gk' - gk' denote ~ , -module structure of VkX,. We set 9k: ----co ~gk'~, and we let ~x: ^x,~ ~~c the Lie algebra homomorph i sm defined by the condition that ~ o ~ x ~ - ~ , . We choose x ^ x X ~Gk, and (by universality) a Lie algebra automorphism ~ of ~ , , such that

(c27)

We then have, from Lemma (CI2), from (C25) , and from the above, that

(c 8)

where ~=~-Io'2. We define J C G k , ,^x as in w (but now for our more general A). I f A is a Classical Cartan matrix (so corresponding to a simple Lie algebra), we

have defined the adjoint group G,a,.~k,(A ) in w 8. More generally, if A is the finite direct sum of the classical Cartan matrices A~-, i~I0, we let G~d,~ k, be the direct product group

G~I ~k' (A) = I ' [ Gad ~k (A/.). , " ~ I , , '

Also, as in w 8 (but for our more general A) we construct the group G~a(A ) and the

homomorphism ~ ' : G~a(A) --+ Gad "~k (A). We let J~----(I)'oAd(J), where ^x

Ad : G k, --+ G~a(~, )

is the adjoint representation. We let .f~C f f be the subgroup of all elements whose reduction s o d t, is in the unipotent radical of the Borel subgroup corresponding to b'.

We let .~~ and for j > I , we let .f~Jl be the subgroup of all elements in Jv

(and in J ) whose reduction s o d t j, is the identity. As in w 18 (but again, for our more general A) we can define the Lie algebras gr over the truncated power series ring ~9~, j > o .

We let '~)C i'~ be the subalgebra of all elements whose reduction s o d t i is zero,

for j_> ~, and we let .~;(~ _ iv. For subgroups H~, H 2 of a group G, we let [H~, H~] C G denote the subgroup generated by the commutators h~h~h~h2 -~, h~eH~, h2eI-~. By

considering the adjoint action of f f /~ i+~ l on Or we see that f f /~ i+~ l is a Lie

group, and a linear algebraic group, with Lie algebra i/i~ +~1. By considering Lie algebras, it then follows that there are integer p > o , q>o , such that

p-times

q-times

r..., .]]=

I3o H O W A R D G A R L A N D

It follows that for each j ~ o , the group ~ J ) i s a characteristic subgroup of J~.

Hence, Ad Z-~o~ induces an automorphism of ~j '~i+a), for each j > o . Since

. .~/~j t-1) is algebraic, it follows that, regarding I?' (the Cartan subalgebra of fl') as a

subalgebra of i/i~ +1), we have that Ad X- ~o~(tl') is conjugate to I?' by an element

of ff/ .~j+l). By a simple argument one can then pass to the limit as j ~ o c , and show _ _

that I?' and Ad Z-lo 7{ I?') are conjugate in i by an element of J (to pass from j+ 11

to ~"/~'u , it suffices to conjugate by an element of ~u~+~)/2(J+~hl~u /. Thus, multi-

plying X by a suitable element of ~ we may assume (setting 7 = Ad Z -a o•

(C29) "~(T) --~t

and hence we may assume in (C28) that we also have

(C3o)

From (C29) , we have the map ~ : I?'-+I?', which induces a dual map (still denoted by ~) on (I?')*, the k'-dual space of b'. From (C29) , and from our earlier observation

that i~ / is a characteristic subalgebra of~, it follows that ~ (on ([?')*) leaves the set of roots A (and the set of simple roots, relative to our fixed order on A) invariant. Hence ~ induces an isomorphism of the root system A, and this isomorphism extends to an automorphism p of g. Of course, by ~%-linearity, p extends to an automorphism of 9~ ~, (which we also denote by p). Then p-ao~ still satisfies (C29), and induces the identity on t?. We let H 1 , . . . , H t ( t = d i m I)) denote the simple coroots in I? (we had fixed an order on the roots A). Also, let

.....

be a Chevalley basis of g. Thus, in particular, for each 0~eA, E~ is a non-zero vector in the e root space and [E~, E_~]----H~, the coroot corresponding to ~. In particular, for each simple root ~i, i = I , . . . , 4 we let E i = E ~ , F i = E ~, and we then have [E~, Fi]=Hi. , i = I , . . . , g. We also have, as we just noted

(C3I) 0 - - 1 o ~ ( I ~ ) = I - I / . , i z I , . . . , t ~

Also, since p - ' o ~ (in place of'~) satisfies (C29), it induces a map on'~u/'~ ), and (C3 I) implies that this induced map leaves each root space invariant. Hence for each eeA+,

we have

p-lo~(E~) = ~ E ~ , e~e ~)',

(9*( C d)C c~k, ) denoting the group of units in the ring 0 of formal power series with

coefficients in k'. But then (C3I) implies for ~eA_

306

But for

and so

THE A R I T H M E T I C T H E O R Y OF LO O P GROUPS

[a=E=, a_ =E_ =] =0 -~ o g(H=)---- H=,

a._==a~-~; in particular, a_=~*. We also have (=eA+)

~- ~o'~( tE ~) = tz2 ~ E_~,

To summarize:

Lemma (C32). - - The map p ~o~ on gk' satisfies the following:

(i) p-~o~(b')=I) ' , and 0-ao~ restricted to I?' is the identity. (ii) For each ~eA+, we have

~-~ o'S(E_ =) = ~;- ~E . . . .

p-~ o'~(tE_ =)= t-:j~E_ ~,

o-~ o'~(t-lE~) = t-~%E~,

We may lift p to an automorphism ~ of fi~,

~ - ' o~ (h )=h , for all he~ (where ~ " -~ = ~ k (I)), with

131

and we will now show that A C ~ C

~k: flk ->9k being the natural projection). To do this, it suffices to assume g simple (and hence absolutely simple, since g is assumed to be split). But then we have ^c ~c gk'= g~,| and ~ is defined by setting ~(~,s)=(p(~),s), ~eg~,, sek'. Moreover, for 0~eA+

A - - 1 o~(H~, o ) = ~-lo~([(E~, o), (E_s, o)]) =[(%E~, o), (.21I~ ~ , o)]=(H~, o),

since ~ is a unit. Also, for eeA+

( ~-:o~ H ~ , (~.~) =~-'o~([(tE_~, o), (t-~E~, o)])

= [(tx21E_~, o), (t-lv~E~, o)]

where Res=residue, = ( H _ ~ , - - R e s ( t z ; l d(t-lv~)) ( - ~ ) ,

---- ( ' - , , & ) ,

and thus we have proved (also see (C3o))

(C33) ~-lo~(h) =h , for all h ~ .

But then it follows that if Xe(~)* where, by definition, we set

~(X) (h) = X(~(h)), h ~ .

is dominant integral, ~(X) is dominant integral,

307


Also, since p on [~ induces an automorphism of the root system A, it follows that ~ on ~',

and hence (by C33), ~ on ~' has finite order. As x varies over ~, ~ (which depends on x) varies over a finite group. We let {~0, X l , . - - , Xq} denote the set of distinct elements in the orbit of X under this group and we set

(C34) "~= 0_<3]].< qV~. G k, and the Lie algebra ~, admit diagonal actions on V, and We note that the group ^x

also that if

= (Ad Z x) o

(see (C28), and the subsequent definition of ~) then ~ induces a bijection ~x of V, and that this ~x satisfies (C26) (but with V~, replaced by V----we note that if the Cartan matrix A has no non-trivial automorphisms, then V=V~,) .

Now, as earlier in this appendix, we let I denote a semi-simple Lie algebra which is defined over 4 and which splits over 4 ' . Let L denote the group of ~s points of the simply connected, linear algebraic group with Lie algebra I (more precisely, we should here replace I by its tensor product with an algebraic closure of 4 ) - We let G%, denote the group of 4 , - r a t iona l points of this algebraic group, and for • we also let • denote the corresponding automorphism of Ga%,. We then construct the fibered product

C G k, • G.~k, , X dominant integral,

as in w I2 (see ( I~ . I4)) , but now for our more general A. Then ~x induces an automorphism ~ of Gk, , by conjugation. We let

• h )=(~(g) , • (g, h)eEX(G~,k,),

and thus obtain a lift of our Galois action on G.~k, to Ex(G.~k). We let I', C EZ(Gs@ denote the Galois fixed points. Then the projection onto the second factor

induces (by restriction) a group homomorphism

L-~L, and by Hilbert 's theorem 9 o, this latter homomorphism is suriective, and hence yields a central extension of L.

308

BIBLIOGRAPHY

[I] A. BOREL, Introduction aux groupes arithmdtiques, Paris, Hermann, I969. [2] A. BOR~.L, Admissible representations of a semi-simple group over a local field with vectors fixed under an

Iwahori subgroup, Inventiones math., 155 (i976), 233-259. [3] N. BOURBAKI, Groupes et alg~bres de Lie, chap. 4, 5 et 6, Paris, Hermann, 1968. [4] F. BRUHAT et J. TITS, Groupes rdducdfs sur un corps local, Publ. Math. LH.E.S., 41 (1972), 1-251. [5] F. BRUHAT et J. TITS, Groupes algdbriques simples sur un corps local, in the Proceedings of a Conference on

Local Fields, held at Driebergen (The Netherlands), Edited by T. A. SP~INGEI~, New York, Springer- Verlag, i967, pp. 23-36.

[6] H. GARLAI~D, Dedekind's ~-function and the cohomology of infinite dimensional Lie algebras, Proc. Nat. Acad. Sci. (U.S.A.), 79. (i975) , 2493-2495-

[7] H. GARLAND, The arithmetic theory of loop algebras, f . Algebra, 53 (1978), 48o-551. [8] H. GARLAND and J. LEI'OWSKY, Lie algebra homology and the Macdonald-Kac formulas, Inventiones math.,

154 (I976), 37-76. [9] N. IWAHORI and H. MATSU~OTO, On some Bruhat decomposition and the structure of the Hecke rings of

p-adic Chevalley groups, Publ. Math. LH.E.S., 25 (1965), 237-28o. [xo] V. G. KAC, Simple irreducible graded Lie algebras of finite growth (in Russian), Izv. Akad. Nauk. SSSR,

32 (I968), I323-I367; English translation: Math. USSR-Izvest~ia, 9. (i968) ' 1271_i3ii. [I I] V. G. KAC, Infinite-dimensional Lie algebras and Dedekind's ~q-function (in Russian), Funkt. Anal. i Ego

Prilozhenlya, 8 (I974), 77-78; English translation: Functional Analysis and its Applications, 8 (I974) , 68-7o. [12] R. MARCUSON, Tits systems in generalized nonadjoint Chevalley groups, J. Algebra, 34 (I975) , 84-96. [I3] H. MATSUMOTO, Sur les sous-groupes arithm~tiques des groupes semi-simples d6ploy6s, Ann. Scient. ]~e.

Norm. Sup., 2 (4th series) (I969) , 1-62. [I4] J. MIL~OR, Introduction to algebraic K-theory, Princeton, Princeton University Press, i97i. [15] R. V. MooDY, A new class of Lie algebras, J. Algebra, 10 (I968), 211-23 o. [i6] K. V. MooDY, Euclidean Lie algebras, Can. J. Math., 21 (I969), I432-I454. [17] R. V. MOODY and K. L. T~o, Tits' systems with crystallographic Weyl groups, J. Algebra, 9.1 (i972),

178-i9 o. [18] C. C. MOORE, Group extensions ofp-adic and adelic linear groups, Publ. Math. LH.E.S., 35 (1968), 157-222. [19] J.-P. SEe.RE, AlgObres de Lie seml-simples complexes, New York, Benjamin, I966. [2o] R. STEINBERG, Gdn~rateurs, relations et rev~tements de groupes algdbriques, in Colloque sur la TMorie des Groupes

Algdbriques, held in Brussels, I962 , Paris, Gauthier-Villars, 1962 , pp. 113-127. [21] R. STEINBEI~G, Lectures on Chevalley groups, Yale University mimeographed notes, 1967.

Department of Mathematics Yale University Box 2155 , Yale Station, New Haven, Connecticut o652o

Manuscrit refu le 27 juillet 1977, rgvisg le 26 fgvrier 1979.

309

LIST OF NOTATIONS

o. R, -s R* (also, see the discussion at the end of w o, concerning the use of "k") I. C, R, Q, Z

after (I .8): B2(a, V), Z2(a, V), H2(a, V) (where a is a Lie algebra over a field k); a t (for fEZ2(a , V))

2. ~, kit, t-l], after (2.I): ~, c after (2.2): v, (,) (on fl), g) after (2.i9): D, A, g~ (for ~ A ) , Ha, E:~ after (2.25): (,) (on D*) after (2.35): hi

3. B, gl(B), ei, f i , hi after (3.2): g(B), b, ge(B), be(B), be(B) * , a 1 . . . . , a[, A(B), A~(B), R, % D 1 . . . . . D l after (3-3): r~ (in Aut(De(B)*)), W=W(B)

after Proposition (3.3): A, A, A, A• I?, Ei, Fi, Hi, ~1, . . . , at , s0, D, b (as the span of D) in (3.8): r: in (3.9): ~b

4. in (4. I): r in (4.2): tie after (4.2):

in (4.3): A+(A) after (4.3): AW(~'), AI('~), AW,+(A), Ar.=t=(p~) after (4.5): Do, ~a, ~i(h), hi in (4.6): (I) after (4.8): U• ~=t=, U E, gZ(~), gz(A) ' gR(~) ' gl~(A), gl~' ~Z, llz~(A), u~, ll~(A), U~, DI~(A), ~I~(A), ~ , • ~

5. g~(X)i, ~.~ in (5-2): I I after (5.6): g R ( ) , g~

after (5-7): U~, Utt~e, ~+ in Remark (5.8): (-5Pt~ =)R[[t , t - l]] after (5.9): r after (5.12): 0

- X VvZ, a , a~ , V~.z 6. tt+(B), pe, ~(a) (for a Lie algebra a over a field k), D, M(k), ~e, VMIX), V x, v0, O//z(A), VZ, after (6.1): V~

in (6.2): Vv~ .I~

after (6.3): bz(A), gl, D~(X), g~ after (6.7): flz(A), gI~() , D~(X), DR() , after (6.io): rr ~z x

after (6.i2): ~b after (6.13): hb, ri (in Aut(De(A)))

310


7" before (7.I): k after (7.I): ~//I~(A), 7~XR, adi~() , ~(4-ai(S ) after (7.3): a2/b(S) after (7.6): @ in (7.9'): wX in (7-r4): Za(s) after (7.I7): ~ , tg* in (7.19): XaX((~(t))= Xa((~(t))

Definition (7.2i): G = G ~ = G ~ ( A ) in (7.22): wX~(~(t))= wee(or(t)) , h~(~(t))=h~(~(t)) in (7.23): wXa(s)=Wa(S), hZa(s)=ha(s) Definition (7.24): or

8. in (8.2): ~ ( a ( t ) ) after (8.7): .oq ~, .~ea(a(t))

after (8.I2): Gad. Xa=Gad.s Gad(A) after (8.13): ~)'

9- after Lemma (9.1): { , }, *, x* (for xGg(A))

after (9-3): u* (for u~q/(g(,~))) xo. GLzk=G,2,k(A), ~e~((r), w~(6), h~(c~), G, E(GLak), ze(a), q~e

in Lemma (IO.I): LiP8 II . U e , we((~), he((~), T e, M e

in Proposition (11. I): U e , U e_ after Remarks ( I I .~) : U~, U+, U_ , T, ra, ha after ( II .3) : T x, U x UX+ U x after (I 1.4): Be, Ne after (lX.5): M~ after ( I I .7) : v, v e, v~, "~u, Ma, "~, Ua , M~, T* in Lemma (11.8): U~,~

12. dp([~), coherently ordered basis, J v after (I~.4): gh in ( I2 . I I ) : co after (12.I3): 7vt, r:a, EX(Gs , G ~ k

after (I2. I4): q~i = q~X, i = I, 2, rr

in (I2.15): bl(~,~) after (12.15): h~(a) in (I~.I7): ba(o',~') after (12.I7): M, S(k*,M) after (12.18): S~ *, M) in ( , ~ . ~ o ) : c~( , ) after Theorem (12.26): pX

13 . in (13.1'): s~.*+v after (~3. V): definition of "equipoltentes" after (13.i ' ) : N*, vp in (13.3): r~ in (~3.4): n.tz* after (13.5): A, a~,mVa~,m, Y,, g, Oaa, m, r~, m in Proposition (13.6): p(n), U~ after Proposition (13.6): T~ after (t3" 13): A

after ( I3 .16 ' ) : f~ , U f ~ = U ~ , Uq_,f 1

311


in Remark (i3. x 7): C

after (13.x9): W, I~ )', S I4. in Proposition (I4.8): N

after Theorem (I 4. Io): X H k = H k 15. in (I 5 . I) : ),i

after (I5.1): ~, ~r , 7~), in Theorem (I5.9): 7;()'1, )'~) in ( I5 . I I ) : c~((r 1, (~) after (I 5. I2); lFe, ~k after (I 5. I8): H~

x6. after (16.I): { , } Lemma (I6.3): tFa

Definition (x6.7): ~ = ~'~ after ( I6. i2) : Hk,+, Hk, 0

17. before ( iT. i ) : D~(A)* after ( I7 . I ) : ~o

after ( i7.2 ') : ~) after (I7.9): e h

after Lemma (x7.x4): J, *~fj, ~ = r k z

I8. after Lemma (x8.2): ,dJ) ~(J) o 'U, x U , r U after (18.3): ~(J), ~(J) after (I8.5): ~j after (18.7): r~ ~ Gk, Uk, d u , ~ , g~(A), U ~ after (x8.8): U~, U ~ in Definition (I8.I5): 0 ~ , ~ , or 9 in Remark (I8. I7): to, u , J[[t]]

x 9. before Definition (x 9. I): ~o in Definition (~9.x): Ha in Definition (r9.2): ~ after Definition (I9.2): ~o before (19.5): g9

20. before Proposition (2o .Q:) , i

before Lemma (20.8): G~' e before Lemma (2o. xo): ~ , Ho z, H~R=Ho z ' l l , S R o = $ ~ ' R , "~' "" I~0 = F 0 , J ~ = J u , H ~ , + = H k , + , K ~ =

2 I. before (2 I . x): qb w in (21.I): <r in (21.2): bj before Lemma (21.3): W~ after (21.5): c%, mj, m after (2i.x5): W~, WF, PF

after (OI.~O): ~ t ' ~R(r) Appendix I. before Lemma (A. x): ~)R(), g R ( ) , q/a(S), ~a(S), ~(s)

in Lemma (A.4): g = T ( b, a)

312

Documents

The arithmetic theory of loop groups