on injective modules for infinitesimal algebraic groups, i

ON INJECTIVE MODULES FOR INFINITESIMALALGEBRAIC GROUPS, I

EDWARD T. CLINE, BRIAN J. PARSHALL AND LEONARD L. SCOTT

Let G be a connected, semisimple algebraic group defined over an algebraicallyclosed field k of characteristic p > 0. We assume that G is defined and split over theprime field k0. In general, for any positive integer r, and any affine fc-group schemeH defined over k0, the r-th infinitesimal subgroup scheme Hr of H is defined to bethe (scheme-theoretic) kernel of the r-th power of the Frobenius morphism o\H-> H.In this paper we study injective modules for Gr and various other important subgroupschemes of G. Fix a maximal torus T of G defined and split over k0, and let H be anyconnected (reduced) T-stable closed subgroup of G. Form the subgroup scheme THr,the pullback of T under ar on TH:

THr

TH >TH

A 77/r-module is by definition a rational module in the usual sense of group schemes,cf. [3] or [7]. Equivalently, these are the hy(i/r)-7-modules studied by Jantzen [11](where hy (Hr) is the hyperalgebra of Hr [3]).

Let O be the root system of T in G. For ae<D, let Ua be the correspondingone-dimensional T-root subgroup of G and set Ua r = (Ua)r. Then we have thefollowing.

MAIN THEOREM. Let H be a connected {reduced) T-stable subgroup of G. Afinite-dimensional THr-module M is injective if and only if the restriction M\u isinjective for each root ae<X> with Ua £ H.

The theorem is false in general if M is infinite dimensional (cf. Example (3.2)below). We remark that it is not possible even to formulate an analogous result forrational G-modules (necessarily allowing M to be infinite dimensional and takingH = G), since the restriction of any non-zero rational G-module to a closed, connected(reduced) unipotent subgroup other than {1} is never injective [4]. It would, however,be possible to formulate an analogue using the various subgroups Ga — <f/a, f/_a> oftype Ay, and we leave the validity of such a possible characterization of rationallyinjective G-modules as an open problem.

It is well established, cf. [11,12,19], that the structures of the G- and TGr- injectivesare intimately related to the problem of determining a ' Weyl character formula incharacteristic/?' for the irreducible rational G-modules. In particular, knowledge of

Received 22 November 1983; revised 19 June 1984.

1980 Mathematics Subject Classification 20G40.

This research was supported by the National Science Foundation.

J. London Math Soc. (2) 31 (1985) 277-291

278 EDWARD T. CLINE, BRIAN J. PARSHALL AND LEONARD L. SCOTT

the TGi-injectives would settle the Lusztig conjecture [18], which proposes such aformula for all large primes p . This is discussed in more detail in the appendix but,briefly, the Lusztig conjecture, when reformulated in terms of rG^-injectives, associatesone of Jantzen's generic patterns [13] to each injective indecomposable r^-module,and describes the structure of the injective up to a filtration whose factors are inducedmodules X \TB\- Here B is a Borel subgroup of G containing rand X is a one-dimensionalTBX-module. The first author intends in a later paper to describe certain canonicallydefined sections of these injectives which (conjecturally) account for the contributionof each alcove to the ^-polynomials discussed by Lusztig [17].

Each r(/r-module M which is filtered by submodules with sections of the formX ITS' (f°r one-dimensional X) is automatically 77^-injective, where B° is the Borelsubgroup containing T which is opposite to B. In fact, having such a filtration isequivalent to being rf^-injective [11], cf. also Proposition (1.5.1) below. Oneconsequence (cf. §4) of the main theorem is that, given a rGy-module M, the collectionof parabolic subgroups P containing T with M \TP not injective defines a subcomplexof the Coxeter complex of G which has a connected complement. It seems likely thatthis associated complex will play a useful role in the study of modules M havingfiliations with induced module factors as above, such as filtered submodules ofinjectives. The study of such modules represents only one of several possibleapproaches, some indeed under active consideration by the authors, to the charact-eristic/) Weyl character formula problem. However, the possibility of mastering a largeclass of important non-completely-reducible modules is in itself appealing, and it islargely in this spirit that this investigation is undertaken.

The first author thanks Vito and Kathy Adamonis for special support andencouragement during the writing of this paper. The second and third authors wouldlike to thank Clark University for its support and accommodation. The authors arealso grateful to the referee and to Professor J. E. Humphreys for suggesting numerousimprovements in the manuscript.

1. Preliminaries

(1.1) NOTATION. The following standard notation will be used for the semisimplegroup G. For convenience we shall assume that G is simply connected. The readermay easily verify that the results of this paper remain valid in the non-simply-connected(or even reductive) case.

T fixed maximal split torusO root system of T in G<X>+(respectively <D~) fixed system of positive (respectively negative) rootsIT = {al 5 . . . ,aj simple roots defined by O+

W = <5ai,..., sa > Weyl group of G, a Coxeter group with involutory generatorssa, for a e IT /

<, > W-invariant, symmetric inner product on ROA = X*(T) weight lattice on T, spanned by fundamental dominant weights

©!,...,©i (recall that (coâf) = Sip where ctf = 2a//<a/ ,a /»A+ dominant weights: the non-negative integral combinations of

(ax,...,col

A+ r-restricted dominant weights (consisting of those AeA+

satisfying 0 ^ <A, aj> < pr, for aêll)

ON INJECTIVE MODULES FOR INFINITESIMAL ALGEBRAIC GROUPS, I 279

^ partial ordering on A: X ̂ n if and only if A—^ is a sum ofpositive roots

p | S a = w1 + ...+coi (see above for cot)oceO+

X \-> X* opposition involution on A: X* = — wo(X)w0 longest word in WUx T-root subgroup of G associated with a e OIrrespectively U~) the maximal connected unipotent subgroup II(X6(I)+ Ua

(respectively n ^ d ) - Ua) associated with O+ (respectively O~)B+ = U+T Borel subgroup associated with <X>+

B~ = U~T Borel subgroup opposite to B+B = UT some fixed Borel subgroup containing T. We identify the

rational character group X*(B) with X*(T), and for X eX*(B)we let X also stand for the corresponding one-dimensional(rational) 2?-module. Usually B = B~

B° = U°T Borel subgroup containing T and opposite to B

A discussion of the basic representation theory of G is contained in [3,6]. ForX e A+, let S(X) be a corresponding irreducible (rational) (7-module of high weight X.Up to isomorphism, these are the distinct irreducible G-modules. The restrictions ofthe S(X), for XeA+, to Gr give the distinct irreducible Gy-modules. It follows easilythat the irreducible rGr-modules are of the form pr0 ® S(r), for xe A+, OeA. (Here,as elsewhere, pr 6 denotes the one-dimensional rG>-module defined by the rationalcharacter pr6 on TGr.) For XeA, write X=pr9+r, for OeA, r e A;!", and setS(X; r)= pr6® S{x). In this way the irreducible rGr-modules are parametrized by theentire weight lattice A. (For AeA+, we shall often write S(X) for S(X;r).) Similarremarks apply to BGr; cf. [5].

For XeA, we let Q(X;r) denote the injective hull of the irreducible rGr-moduleS(X;r) in the category of rational rGy-modules.

For an arbitrary aflfine algebraic ^-group scheme H, let MH be the category ofrational //-modules. If K is a closed subgroup scheme, the restriction functorMH -> MK(Ki-> V\K) admits a right adjoint

MK—>MH{Wv—> W\» = W\»),

called induction. More generally, if / : K -*• H is an arbitrary morphism of affinealgebraic A;-group schemes, we have restriction f*: MH -* MK and inductionf*:MK -> MH functors. For a discussion of these matters, see [3,§ 1].

(1.2) Integrals. If H is an affine algebraic group scheme defined over k0, we lethy(//) denote its hyperalgebra, cf. [3, §6] for example. If FeObtMj,) , then Fis ofcourse a hy (H)-module in a natural way.

When H is infinitesimal [7, II, §4.7], hy(//) is a finite-dimensional Hopf algebraover k, so has a unique (up to scalar factor) left integral Jw e hy (//). Recall that thedefining property of ]# is that it is a non-zero element of hy (//) satisfying

r \ = e(X) f ,JH JH

where e:hy (//) -• k is the counit. Also, hy (H) is a Frobenius algebra, therefore thenotions of an injective module and a projective module coincide. We refer to [6] for

M


details of the above. If H is in addition unipotent, then every injective hy (H)-moduleKis free and, if {yj is a hy(i/)-basis of V, then {{//tfj is a A>basis for VH.

Observe also that for H infinitesimal unipotent, if KeObCM^), for veV, and\H v # 0, then hy (H) v is a free hy (//)-module. (Note that the natural surjectionhy(/ / ) -+ hy(H)v is injective on the socle of hy(//) , hence it is an isomorphism.)

To give an example of \H, take H = £/+. Then

[ = I I , r , for some fixed order on O+,

in the notation of [3, (5.1)].

(1.3) Serre duality.} Fix B = B~. For XeA,r ^ 1, we If, we define

Here WB = wBw'1. We write Ak for Ax(\;r) when r is fixed by the context; thus,Ax(w'>r) = Wj4w-ix> where the superscript denotes conjugation of the module (lettingthe group act through conjugation by w"1). It is verified by the universal mappingproperty of induction that as a rGy-module, Ax has irreducible socle isomorphic toS(X;r). Also, for A* the dual of Ax, we have that

Af £ hy(TGr)®hy{TBr)-X s hy(£/+)(g)-A

(cf. Remark (1.4) below). Hence, projection onto the ( — X + 2(pr — l)/?)-weight spaceof A*, namely the space j(/+® — X, defines a !Ti?r-homomorphism which lifts to a7Yjr-module map A* -> A_x+2(pr_l)p. It is clearly injective on the £/+-socle (which is$u+A*), and so is an isomorphism since both sides have the same dimension. (Oneargument for this is that Ax \v+ = X |2 \

u? = @(U?), the coordinate ring of £/+, by [5,(4.1)]!)

More generally, for V a finite dimensional 77?r-module, we have a Serre duality

T T

for B = B~, easily proved from the one-dimensional case by induction on thedimension of V. We leave the remaining details to the reader.

If P = RU(P) • L is a parabolic subgroup of G containing T, there is an obviousanalogue of (1.3.1) obtained by replacing G by P and p by the half-sum of the positiveroots in the Levi factor of P. Note also for G = SL2 that (1.3.1) says that

Af ~ A/ I jj ;-— / I n . 1

where sa r-X = — X + (pr— l )a .Finally we record, for general G, that Ax has formal character given by

\—evT* \—e-p

r<*

<xe<D+ * e

which follows from the T-isomorphism >1* ^ hy(t/+)® —A (cf. also (1.4.1)—(1.4.3)below).

f The results of (1.3), (1.4) were obtained by the authors while in residence at the Institute for AdvancedStudy during the winter term, 1979. Analogous results have also been proved by Kempf [15].


(1.4) REMARK. When ATisa closed subgroup scheme of an infinitesimal algebraicgroup scheme H, there is a way to define a 'production' functor MK -> MH whichis a left, rather than a right, adjoint to restriction. For Ve Ob (MK), the associatedproduced module is

essentially as used above. This is the analogue of the classical notion of induction inthe theory of Lie algebras. Humphreys [9] used this Lie algebra approach to definea family of modules Zx for the restricted Lie algebra of G. This was later generalizedby Jantzen [11,12] to the study of hy (Gr)-r-modules. We generally prefer to use theAx instead of the Zx, and to use induction (which is natural for rational modules forall group schemes) rather than production. However, for the convenience of the readerwe note the following results, the first of which is just the definition of Z^ for X e X*(T):

(1.4.1) Zk s hy(Gr) <8>hy(B+) A, as a hy(Gr)-r-module;

(1.4.2) ZA £ A.x(w0; r)* s Ax_2ipr_1)p(w0; r) as rGr-modules;

(1.4.3) [Zk ] = [Ax ] in the Grothendieck group of Tor rGr-modules (where an easyargument with maximal weights shows that the latter Grothendieck group injects intothe former);

(1.4.4) Zx is a rank-one free hy(£/;:)-module, generated by a vector of T-weight X,and Ax is a rank-one free hy(C/+)-module with socle X (alternatively, Ax is thert/J-injective hull o

(1.4.5) Zx has a unique rGr-irreducible quotient module S(X; r), which is the uniqueirreducible submodule of Ax-

(1.5) Filtrations. The following result is essentially due to Jantzen [11]. AgainB = B~.

(1.5.1) PROPOSITION. Let M be a finite-dimensional TGr-module. Then M\TB+ isinjective if and only ifM has a filtration of TGr-submodules whose successive quotientsare each isomorphic to an Ak.

Proof. If M has a filtration with each section isomorphic to an Ax then M\TB+is injective by (1.4.4). For the converse, let X be the largest T-weight occurring in Mwith non-zero multiplicity. Then we clearly have a non-zero 77?r-homomorphismM->A, hence a non-zero 7I(jr-homomorphism (f>\M-*• X\TGr ^ Ak. Let D e M b e aA-weight vector with <f>(v) # 0. Certainly, v is fixed by Uf, so can be used to definea r^!"-map X -*• M with image kv. Since M\TB+ is injective, this extends to a map ofthe TB$ injective hull X (g) 0{Ut) of X to M which, composed with <f), gives a non-zeroTB+-map to Ax'.

M

Since X is by definition the socle of its injective hull, this map to Ax is injective, hencesurjective by dimension considerations. Thus, the rGr-module map ^ is a splitsurjection as a 7 ^ - m a p , and the proof may now be completed by induction on thedimension of M.


(1.5.2) REMARK. If Mis a i?G>-module with M \TB+ injective, then the above proofshows that M has a filtration with successive quotients isomorphic to a — X ||Gr. Forp^2h — 2 (where h is the Coxeter number of G), each injective Q(X; r), for .̂eAjt",has a G-, hence i?<jr-module structure [11,4.5], and hence admits such a filtration.

(1.5.3) The following further generalization of (1.5.1) is required for the proof ofthe main theorem in its full generality.

PROPOSITION. Let Hbea connected T-stable closed, reduced subgroup ofG. Assumethat notation is chosen so that B+ OH is a Borel subgroup of H. Let M be a

finite-dimensional THr-module. Then M | T ( B + n / / ) is injective if and only if M has afiltration whose successive quotients are isomorphic to ^\T"B-(\H) >for ÊX*{T).

The proof follows that of Proposition (1.5.1).

We note the following result which shows that the Z^ and the A^ work welltogether.

(1.5.4) PROPOSITION. For X, // e A, we have

Ext" (7 A

Also, E\t^G (AM, Zx) has the same value.

Proof This follows from [3, (4.3)], (1.5.1) and (1.4.4).

The relevance of this result to filtrations is the following.

(1.5.5) COROLLARY. Suppose that M is a finite-dimensional TGr-module which hasa filtration by submodules with factors of the form Ak. Then the multiplicity of agiven Ax in the filtration is dim Hom r G {ZhM) = dim HomTB+ (X, M). Also

,M) = O forn>0.

Proof This is clear from (1.5.4), (1.4.1).

Finally, for later use in the appendix we record the following result.

(1.5.6) PROPOSITION. For n > 0, X,fieA, we have that Ext?.G (AÂft) = 0 unlessX > n.

Proof By [3,4.3], we have that Ext£Gr (Ax, AM) s Hn(U~; Af ® fi)T'. Since theweights v in A* ®M satisfy v ^ — X+pi, the desired conclusion follows from an easyweight argument on the normalized cochains.

2. Reduction to Borel subgroups

The following theorem is of course a consequence of the main result, but we needit as an intermediate step in the proof of the latter.


(2.1) THEOREM. Let Hbea connected {reduced) T-s table closed subgroup ofG, andlet M be a finite-dimensional THr-module. Assume that notation is chosen so that B+ 0 His a Borel subgroup of Hand choose a Borel subgroup B' ofG containing Tso that B' OHis opposite toB+OH (that is, B+ 0 B' 0 H = RU(H) • T). Then M is injective if and onlyi/M\T(B+ n H)r and M \T(B- n H)r are injective.

Proof For simplicity we shall just give the proof in the case when H = G. Thegeneral case may be treated similarly using (1.5.3).

If Mis injective, so are M\TB+ and M\TB- by [3,4.2] since TGr/TB+ and TGr/TB~are both affine (isomorphic, in fact, to U^, and U? respectively [5, §6]).

Conversely, assume that M\TB+ and M\TB- are injective. Then by (1.5.1) and(1.3.1) the module M®M* has a filtration by TCy-submodules with successivequotients isomorphic to r(?r-modules of the form Ax(\;r)®Au(w0;r). The latter,however, are injective on TGr, since for 5 e O b ( M T G ), we have

)®fi®S)\lGB') (by[3, 1.2.4])

by [3, 4.3], (1.5.1), and [3, 2.1]. It follows easily that M ® M* is injective.Finally, note that M ® M* = Endfc(M ), and that the surjective TGr-map

Endfc(Af) ® M • M: <f> ® m i • <f>(m)

splits (by m \—• \M ® m). Thus M is a direct factor of an injective 7Yjr-module, henceis injective.

(2.2) REMARK. One way to phrase the above result, say for H = G, is that, givena TGr-modu\e M, one can detect whether or not W^G,, S ® M) = 0 for allrG>-modules S on the subgroup schemes TB+ and TBj. This is unfortunately not truefor the case of a single rGr-module S. Take, for example, S to be the cokernel of theinclusion k -*• G(G^). Note that ^(TG^ k) # 0, since any T-linear map Lie G -> k givesrise to a central extension of TGX in a natural way [8, 3.11]. Thus, HîTG^ S) # 0 bydimension shifting. However,

t, S) S H*(TB+, k) = 0 = HîTB^, S)

as follows from an elementary weight argument with normalized cochains.

3. The injectivity criterion

We now aim at proving the main theorem. In view of (2.1), we can reduce to thecase in which H is unipotent, H £ B+. We shall choose a normal T-stable subgroupof H in a special way to permit an inductive argument. This will be accomplished bymeans of the following result.

(3.1) LEMMA. The positive roots <J>+ relative to B+ may be listed as <X1,...,OLN SO thatthe following two conditions hold.

(a) For each mÔ, Km = n^Lx Uat is a subgroup normalized by Uam+i.

(b) For each m ^ 0, there is an ordering ^m on the Euclidean space UQ> such that

<*i >mQfori^ m> and am+l < m 0-


Proof. Let Ko = {1}. Then (a) holds vacuously, and so does (b) if we take for ^ 0

an ordering for which the elements of <D~ are all positive.Suppose that m > 0 and we have constructed orderings ^ and subgroups Kj

satisfying (a) and (b) for j < m. Suppose further that we have chosen the roots«i> •••><xm-i s o t n a t t n e r e exists a Borel subgroup Bm_! = TUm^ such that the rootsubgroups contained in Um_x which are positive relative to the standard partialordering ^ (cf. (1.1)) are precisely Uai,..., Uam_x. (Set Bo = B~, Uo = U~.)

Let ^ m be an ordering on IRQ in which the roots associated to Um_x are positive.To choose am, choose a negative root — a m e Q ~ which is simple relative to theordering ^m, and put Um = XmUm_v Let Q , ^ be the set of roots of Um_v Thenwe have that

^m-i n O+ = {al5..., am_J, N ^ m - i ) n O+ = {al5..., am}.

Then (b) is satisfied, as is the required condition on Bm = K m 5 m _ 1 = TUm. Certainly(a) holds with Km = Um0 U+, and the lemma now follows by induction on m.

To prove the main theorem, let M be a finite-dimensional 77/r-module withM\TU injective for Ua c H. We shall show that M is injective by induction on thedimension of H. Using Lemma (3.1), we choose the largest integer m such thatK = H n Km is a proper subgroup of H. We have that H is the semidirect productK X Ua for some root a. Furthermore, there is an ordering ^ on IRQ such that a < 0and p > 0 when Up Q K.

Let J be the intersection of the maximal rA^-submodules of M. Let I be a T-weightof M which is minimal relative to ^ . Then it follows that Mcontains a non-zero A-weightvector v$J. (In fact, y ^ / for any /l-weight vector of Af.) Clearly veMu*<r, since ais negative with respect to ^=. Since M\v is injective and hence a free hy (t/a r)-module,it follows that there is a vector « e M with \ v u = v. Also, since v spans the f/a r-socleof hy (£/a r)«, we see that hy (t/a r ) «n J=0.

Adjusting \H by a non-zero scalar we can assume that

1 - I f •Since u £ / and M is free as a hy (^r)-module by induction, we have that

f u = [ f K = I y # 0.

Thus, M' = hy (//r) M is a free hy (//r)-module and hence an injective 77/r-module.Thus, M decomposes as M = M' 0 M" for some r/^r-submodule M". It now followsby induction on the dimension of M that M" is 77/r-injective. Hence, M is 77/r-injective.

Finally, the converse implication of the theorem follows from [3,4.2].

(3.2) EXAMPLE. The theorem is false if we allow M to be infinite dimensional. Forexample, take p = 2, r = 1, G = SL3, and H = Uax Ua+p (where a and /? are simpleroots). Let M be a vector space with basis {vt, Wj)ijez- Let Xa denote a generator ofhy(â, i) and select Xa+p similarly. Let vt, wt have T-weights $ , <x+ifi respectively.Put

Xa+fi»i = wi+l> X*+0 wi = 0» ^ a »i = WU Xa Wt = 0.

ON INJECTIVE MODULES FOR INFINITESIMAL ALGEBRAIC GROUPS, I

A schematic representation of the module M is

285

a + 0

Clearly, M\Ua i and M\v i are free hy {Ua x)- and hy (C/a+£ J-modules, respectively.Hence, M\TU are injective. However, M has no finite-dimensional 77/^directsummands, hence is not injective.

It seems likely that one could construct analogous examples for/? > 2, by allowinglarger intersections of the modules hy(C/a+îl)ui and hy(C/ail)wi+1. Similarly, oneshould be able to give examples with characteristic 0, using the universal envelopingalgebra and phrasing everything in terms of of free modules (though such exampleswould not be finitely generated). We have not, however, explored these possibilitiesin any detail.

4. The Coxeter complex of a module

Let G be a semisimple algebraic group and T a maximal split torus as in § 1. Recallthat associated to T there is an abstract finite simplicial complex %> defined as follows.The vertices of # are the maximal (proper) parabolic subgroups P of G containingT, and a collection of vertices forms a simplex if and only if the correspondingparabolic subgroups contain a common Borel subgroup. The simplices thereforecorrespond, by taking intersections, to the parabolic subgroups of G containing T.If P is such a parabolic subgroup, we let CP denote the simplex corresponding to P.We call %>, or its geometric realization | # |, the Coxeter complex of G relative to T.See [2] for a detailed discussion (from a slightly different but equivalent point of view).

Given a finite-dimensional rGr-module M, we define the complex of M to be thesubcomplex ^r(M) = #(M) of # consisting of all simplices CP for which M\TP isnot injective. Note that if Q 3 P, then M\TP is injective whenever M\TQ is injectiveby [3,4.2]. Thus, <tf(M) is a subcomplex of i. Note that if <D(M) = {ae<X>: M\Ua r isnot injective}, then, by the main theorem,

<g(M) = {CP:PÛa, for some ae<D(M)}.

In Corollary (4.6) below, we characterize the possible complexes #(M) which can ariseby showing that O(Af) can be taken to be an arbitrary subset of O.

(4.1) THEOREM. Let M be a finite-dimensional TGr-module. Then the following hold.

(a) Each maximal simplex of #(M) has dimension equal to that of %>, namelyrank(G)-l.

is connected.(b) The complement \W\ —

Proof. Let CP be a simplex in #(M). Then by the main theorem, P contains aBorel subgroup B containing T such that M\TB is not injective. It follows that CB

is a simplex of #(M) containing CP as a face, proving (a).To see (b), let B and B' be two Borel subgroups containing T such that M\TB

and M\TB> are injective. It suffices to show that \CB\ and \CB>\ lie in the same


connected component of | # | - 1# (Af ) |. If B,B' are adjacent, then the common wall| Cv | of | CB | and | CB | is contained in | # | - 1 #(M) |, by (a), where P = (B, B'}; hence| CB\ and | CB'\ are in the same component of | # | —|#(M)|. Otherwise, let a be aroot which is simple for the root system of B, but not a root of T in B'. By the maintheorem, M\T{»aB)r is injective, while dim (B n B') > dim (S*B f] B'). The proof is noweasily completed by descending induction on dim (B n B').

(4.2) EXAMPLE. For A e A, let Wkr = (sa: a e n , <A+/>, av> = Omod/?r>IfaeIT,let P = (B-,say, and let St(a, r) denote the Steinberg module (pr- \)coa | ^ l (wherecoa is the fundamental dominant weight corresponding to a). If <A+p, au> = 0 modpr,then X = (pr— l)coa+p: for some p:e A satisfying ( ^ a " ) = 0 mod/;r, and so

X \%pB

r-- s St (a, r)®p.^ 5«St (a, r) ® ^ s (

for some /^'eA. Thus, ^^ ^ ^«ayl+/i'(ja;r) and so ^|T(««B+) is injective. Conversely,if XeA and if Ax |T(«aB+) is injective, then, by (1.5.1) and dimension considerationsit is isomorphic to Ax>(sa;r) for some X'eA. The weight X' is determined by the factthat

However,AX\T(>*B

by [5, (4.2)], so that

Homy/s«j3+\ (A', A

\ = {X\TT%)\™rr\T(saB+)r = (^ \TB;

£ HomTC/_a

/_a r) # 0

for A' = X — (pr — 1) a. Now A |^gi always contains a 7"Pr-irreducible module with high

weight X, and so Hom T l / - (A' , A | y ^ ) ^ 0 for some X' = sa(A) modpr A. Hence, the

uniqueness of A' gives that

A' = .yaA = X-(pr-\)a mod/>r A, <A+/>,av> = 0mod/?rA.

In particular, 5a e Wx r.If Ax\(T

wB+) is injective, it follows from (4.1) that there is a chain

Bo = WB+,...,Bn_x = s « 5 + , B n = B+

of pairwise-adjacent Borel subgroups containing Tsuch that each M \T(B ) is injective.By the above, sa e Wk r. Using induction on n, replacing B+ by s"iB+, Tl by J a ( n ) , Aby sa(X), and p by sx(p), we obtain directly that

(Note that the form < ,> is PF-invariant, and ss ^ = saspsa.) Thus we have thefollowing.

(4.2.1) PROPOSITION. With Xe A and using the above notation,

<$-<g(Ax) = {CP:P 2 wB+,for some we WkyT}.

For our next example, we need the following.

(4.3) PROPOSITION (see [18]). Let Pbeaparabolic subgroup ofG containing T, andwrite P = VL, where V = RU(P) is the unipotent radical and L is a Levi complement.


Set V° = (Ua: U_a £ j/> and put P° = V° • L (the parabolic subgroup 'opposite' to P).Let M be any TLr-module, viewed as a TPr-module by inflation. Then

where here M is viewed (on the right) as a TP°-module by inflation, and the coordinatering (D(V°) is given a TP°r-module structure by letting TLr act by conjugation. Inparticular, M is a TLr-direct summand of

M\TGT IM \TPr

r\TLr-Proof. The product TPr P° is clearly TGr in the sense required by [5,4.1]. (There

are no technical difficulties with epimorphisms of faisceaux here, since all quotientsinvolved have sections.) Since TPr n 77* = TLr, we have that

M\$%\TP.*M\$% (by [5,4.1])

^M®k\VLrr (by [3,1.2.4]),

and it is easy to see that k\%£r is isomorphic to the module O(V°) described in theproposition.

Finally, the 7Zr-module inclusion M -> M% 0(V°) is clearly split.

(4.4) COROLLARY. Let M be a TLr-module as above. ThenMîT\Tpa has afiltrationby submodules whose successive factors are of the form M ® S ® O(V°), where S is anirreducible TLr-module (inflated to 77*.) appearing as often as its multiplicity in 0(Vr).Moreover, the section M (g) &(V°) associated with S = k is the bottom term of thefiltration, and the inclusion of this bottom term is TP^-split. In particular, Mis a TLr-directsummand of M\%,%r\TL .

Proof. First note that

M\TL; ^M®k\Zfrr = M® O(Vr),

as in the proof of (4.3). Now, as before,

r

Observe that ( M ^ ^ ^ ^ ( M ^ ^ j r c , | T p ? ~M®S® ®(V?)for S an irreducible 7Xr-module, by the previous result. Now M is clearly the firstterm of a 7Lr-flltration of M®O(Vr), whose successive quotients are the variousM (x) 5 with multiplicities as described in the statement of the corollary. Moreover,the inclusion M-+M®(9(Vr) is 7Xr-split. Inducing up and applying the aboveobservation, the result follows.

(4.5) EXAMPLE. Let a e O and let # a = {CP :Ua^P} in the notation of (4.1). Oneobtains easily from (4.4) and (4.2) (applied to SL2) that, putting Ja = fc|y£> .

(4.5.1) Ja \v r is injective if and only if P # a.

Thus,

(4.5.2)

More generally, one can show that %>(X |£gr ) = <^0L if X is not a'Steinberg weight'for T(Ua, C/_a>, meaning that if a is simple then <A+/J , au> ^ 0 modpr. In theexceptional case we have <€(X \%fy ) = <€; that is, k \^r is injective.


Next, note that for M, N any finite-dimensional rGy-modules,

(4.5.3) <€{M ®N) = <€{M) U <g{N).

Hence, we obtain for any subset O' £ <b that

(4.5.4) <f( U Ja)= U <€,.ae<t>' aeO'

(4.6) COROLLARY. The possible complexes ^(M), for M a finite-dimensionalTGr-module, are precisely the complexes (Jae(D' #«» where O' ranges over the subsetso/O.

Proof. Apply (4.5) and the remarks preceding (4.1).

(4.7) REMARK. It is interesting to note that, when r = 1, there is a multiplicativeversion of the evident additive relation O(M ®N) = Q>(M) U <D(A0 analogous to(4.5.3). Namely, we have that

(4.7.1) For M, N finite-dimensional rGrmodules, <D(M ®N) = <D(M) 0 ®{N).

To prove this it suffices, by the main theorem, to show that if M\Uix i and N\Uix i arenot injective, then neither is M (x) Nly x injective. Without loss of generality we canassume that M and N are indecomposable, in which case their dimensions are strictlybetween 0 and p. But now p Jf dim (M ® N), so M ® N is free hy (t/ai J-module.

(4.7.2) The above result fails for r> 1, as can be seen by considering thestandard tensor product factorization [6] of the Steinberg module, say for r = 2and G = SL2.

However, it is true (and easy) that for all r,

(4.7.3) <D(Af ® N) c (D(Af) n <&(N), for M, N finite-dimensional rGr-modules.

APPENDIX

One of the points of this appendix is to make explicit the connection between theLusztig conjecture [18] and the structure of injective indecomposable 7'Gr-modules.This connection is implicit in the work of Jantzen [11], whose symmetry results oninjectives seem to have motivated some of Lusztig's work on the Hecke algebra, cf.[17, p. 122]. A second objective of this appendix is to sketch briefly alternative proofsof these results of Jantzen and to give some generalizations. We conclude with anobservation on the ordering of the Ak in filiations of the injectives.

Let 3tf be the system of hyperplanes which define the affine Weyl group Wv of Grelative to the dot action

w X = w(X+p)—p, weWp, XeA,

cf. [11, §1] for a further discussion of Wp. Let f̂0 be the subset of 2tf consisting ofthose hyperplanes H which pass through —p. If C is a chamber for the system Jô,then C contains a unique chamber Ac = A for the system Jf whose closure contains—p. We call A the bottom alcove in C. A weight 0 in A is called a Borel-Weil weightrelative to C (or simply a Borel-Weil weight when it is clear which chamber is underconsideration).

If M is a rational G-module, then for r ^ 0, M{pT) denotes the rational (/-module


obtained by letting G act on M through ar [3, 3.3]. The following result motivatesa precise definition of a generic weight. It generalizes [11, Satz 3.8]. Note that by [5,4.1], each Ax is naturally a iftjv-module, namely A||Gr.

(A. 1) PROPOSITION. Let X be a weight in C, and assume that each BGr-composi-tion factor in Ax has the form pr6® S(z), where 0 is a Borel-Weil weight andxeA+.Let H? denote the n-th right derived functor of induction from MBG to MG, takingGo = {1}. Let we Wsatisfy w-C — C+, the dominant chamber. Then

(a) H?(pr9® S(r)) £ H« (0)(pr) ® S (T) S S(pr(w • 0) + T) is the irreducible G-module of high weight pr(w0) + x only ifn = l(w); otherwise it vanishes.

(b) In addition, Hfi (X) vanishes unless n = l(w) (as in the characteristic zeroBorel-Weil theorem; this is the reason for the terminology' Borel-Weil weight'). In thiscase, the multiplicities of the composition factors of Hl£w) (X) are given in terms of themultiplicities of the composition factors of Ax by the formula

(X): H#w> (0)<*>r> ® S(x)] = [Ax :pr 0 (g) S(r)].

Proof Part (a) follows from the tensor identity [3, 2.3], the fact that, for anyrational 5-module E, HÊf^ ^ H" (£)<*>'> [5, 4.3], and the fact [1] that theDemazure proof of the Borel-Weil theorem holds for the weights in the bottom alcoveofC.

Part (b) follows immediately from the spectral sequence of induction [3, 4.7],together with a simple induction argument (on the length of a composition series forAx) using the long exact sequence for the derived functors of |^G .

We now call a weight X in C generic if the composition factors of Ax satisfy thehypothesis of (A. 1). When C is the dominant chamber C+, the above result isessentially due to Jantzen [11, Satz 3.8] who proved an analogous result for Weylmodules. When combined with the analogue below of Brauer reciprocity, alsoessentially due to Jantzen [12], one obtains the connection between generic weightsand injective indecomposable rGr-modules mentioned in the introduction. For p >̂ 0Jantzen has proved that generic weights exist [11].f

If M is a rational 7Yjr-module which has a filtration whose factors are, say, ofthe form Ax, we write [M:AX] for the multiplicity of Ax as a factor of the filtration.As usual, [M: S(X;r)] denotes the multiplicity of S(X; r) as a composition factor of M.

(A. 2) PROPOSITION. For any pair of weights X,/J, with fie A+, we have that

[Ax:S(M;r)] = [Q(M;r):Ax].

Proof. This follows from (1.5.5), (1.4.3), and the definition of Q(ji,r) as the7(/r-injective hull of S(/i; r).

(A. 3) REMARK. We mention that Jantzen has an Euler-characteristic version [11]of Proposition (A. 1) which applies when 6 is not a Borel-Weil weight. This can alsobe obtained from the point of view of inducing to G from B through BGr, with theEuler characteristic becoming that of the higher derived functors of induction fromBGr to G. However, we omit further details.

t For C — C1", for example, let X =pr v' + k', where v'eA+ satisfies <v',a"> ~&h—\ for all ae<D+ andk'+pe A+. By [11, Lemma 3.4] and (1.4.3), ifpr0+S(z), for reA+, is a 7Gr-composition factor of AX then6eA+. Since, by (1.3.2), pT6+x < X, a straightforward computation shows that 0 is a Borel-Weil weightprovided/) > <v',av> for all ae<D+.

10 JLM 31


In view of (A. 1) and (A. 2), it follows that a determination of the multiplicities ofthe Ax in the Q(ji\ r) would settle the Lusztig conjecture (for/? >̂ 0). The fact that thesemultiplicities occur in a symmetric pattern about the Steinberg weight (p— \)p followseasily from a character calculation [12]. More explicitly, we have the following.

(A. 4) PROPOSITION. Let n= pr6+x, where reA+.Ifwe write

[Q(M;r)]=ic(jii)[Atti)i-l

to describe the factors of a filtration of Q(ji; r) each of whose factors is an Ah then£ ciji^Hi is stable under the action of the Weyl group W = (Wp)preHp_l)p (that is thestabilizer of pr 0+(p—\)p which is centered around the special point prd+(p—\)p).

Proof If fie A+, then S(ji;r) = S(ji) is a G-module, hence an NG(T) Gr-module,and so wQ(ji;r) s Q(M',r) as rGr-modules for all weW. Thus, the formal characterof Q(JI; r) is stable under the action of W. The result now follows for fi by a formalcharacter calculation (checking reflections). The case of a general n is similar, mutatismutandis.

Once one knows the multiplicities of the A^ in the injective modules Q(j,r), thestructure of Q(x; r) can always be described up to a filtration in at least one canonicalway. Pick a total ordering ^ on A comparable with ^ and, for each fie A andfinite-dimensional r<jr-module M, set F^{M) equal to the largest submodule of Mwhose composition factors all have high weights < p..

(A. 5) PROPOSITION. Let M be a finite-dimensional TGr-module. Assume thatM \TB+ is invective. If X is minimal ^n with FX(M) ^ F^M), then Fx(M)/F/J(M) isthe direct sum of[M:Ax] copies o

Proof Assume inductively that the result is true for v -< fi if FV(M) £ F^M). Inparticular, the given F^(M) has a filtration each of whose successive quotients is anAT, hence it has an injective restriction to TB+. Thus, M/F^M) \TB+ is injective andis filtered by the AT. Now use (1.5.6).

We remark that filiations other than these obvious F^ may well give a betterpicture of the injective rGr-modules. An illustration of this will be given by the firstauthor in a future paper.

References

1. H. ANDERSEN, 'Cohomology of line bundles on G/B\ Ann. Sci. Ecole Norm. Sup. 12 (1979) 85-100.2. R. CARTER, Simple groups of Lie type (Wiley, New York 1972).3. E. CLINE, B. PARSHALL and L. SCOTT, 'Cohomology, hyperalgebras, and representations', / . Algebra

63 (1980) 98-123.4. E. CLINE, B. PARSHALL and L. SCOTT, 'Induced modules and affine quotients', Math. Ann. 230 (1977)

1-14.5. E. CLINE, B. PARSHALL and L. SCOTT, 'A Mackey imprimitivity theory for algebraic groups', Math.

Z. 182(1983)447-471.6. E. CLINE, B. PARSHALL and L. SCOTT, 'On the tensor product theorem for algebraic groups',

J. Algebra 63 (1980) 264-267.7. M. DEMAZURE and P. GABRIEL, Groupes algebriques, I (Masson, Paris 1970).8. E. FRIEDLANDER and B. PARSHALL, 'On the cohomology of algebraic and related finite groups', Invent.

Math. 74(1983)85-117.


9. J. HUMPHREYS, 'Modular representations of classical Lie algebras and semisimple groups', J. Algebra19(1971)51-79.

10. J. HUMPHREYS and J. JANTZEN, ' Blocks and indecomposable modules for semisimple algebraic groups',J. Algebra 54 (1978) 494-503.

11. J. C. JANTZEN, 'Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne', J. Reine Angew.Math. 317(1980)157-199.

12. J. C. JANTZEN, ' Ober, Darstellungen hoherer Frobenius-Kerne halbeinfacher albebraischer Gruppen',Math.Z. 164(1979)271-292.

13. J. C. JANTZEN, 'Uber, das Dekompositions verhalten gewisser modularer Darstellungen halbeinfacherGruppen und ihrer Lie-Algebren', J. Algebra 49 (1977) 441-469.

14. J. C. JANTZEN, 'Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren', Math. Z. 140 (1974) 127-149.

15. G. KEMPF, 'Representations of algebraic groups in prime characteristic', Ann. Sci. Ecole Norm. Sup.(4) 14 (1981) 61-76.

16. R. LARSON and M. SWEEDLER, 'An associative orthogonal bilinear form for Hopf algebras', Amer. J.Math. 91 (1969) 75-94.

17. G. LUSZTIG, 'Hecke algebras and Jantzen's generic decomposition patterns', Adv. in Math. 37 (1980)121-164.

18. G. LUSZTIG, Some problems in the representation theory of finite Chevalley groups, Proceedingsof Symposia in Pure Mathematics 37 (American Mathematical Society, Providence 1980)pp. 313-317.

19. L. SCOTT, Representations in characteristic p, Proceedings of Symposia in Pure Mathematics 35(American Mathematical Society, Providence 1980) pp. 319-331.

(Cline) (Parshall and Scott)Department of Mathematics Department of MathematicsClark University University of VirginiaWorcester CharlottesvilleMassachusetts 01610 Virginia 22903USA USA

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