Schubert cells and representation theory

Digital Object Identifier (DOI) 10.1007/s002229900932Invent. math. 137, 461–539 (1999)

Springer-Verlag 1999

Schubert cells and representation theory

Luis Casian?,Robert J. Stanton??

Department of Mathematics, Ohio State University, 231 W 18th Avenue, Columbus,OH 43210-1174, USA

Oblatum 6-VIII-1995 & 23-II-1998 / Published online: 21 May 1999

Introduction

Let G be a linear, real, reductive group, and letP be a parabolic subgroup.The Bruhat decomposition ofG gives a cellular decomposition of the gen-eralized flag manifoldX = G/P. Originating in the work of Schuberton Grassmann manifolds, this cellular decomposition was used by Ehres-mann [Eh] to give a proof of the basis theorem for the integral cohomologyof Grassmannians. Since Ehresmann it has been clear that, for a generalizedflag manifold X, a more detailed understanding of this decomposition bygeneralized Schubert cells would provide a determination of the integral(co-)homology ofX. The aim of this article is to give a representationtheoretic determination of the differentials in the Schubert cell decompos-ition of X and thereby obtain a combinatorial description of the integral(co-)homology ofX. Our primary tool will be the infinite-dimensional rep-resentation theory ofG.

If G andP are complex groups, elementary considerations show that allSchubert cells define non-zero integral homology classes and that none aretorsion. Moreover, the cohomology has the well-known connection to finite-dimensional representation theory ([Ct], [Ko1], [Ko2], [Bt]). For real groupsG andP, even in low dimensions, torsion can be present in the homology ofreal flag manifolds. Thus the major new development in this paper is that theinfinite-dimensional representation theory ofG detects which real Schubertcells define integral, in particular torsion, classes in the (co-)homology.The innovation on the representation-theoretic side that makes possible theconsideration of (co-)homology with integer coefficients is the geometricformulation of representation theory introduced by Beilinson-Bernstein.

To compute homology we use a topological technique that applies tospaces with filtration. LetXp be the union of Schubert cells whose dimension

? Supported in part by NSF DMS-9302702 and the Sloan Foundation?? Supported in part by NSF DMS-9401193, IHES, the CNRS through Universite Louis

Pasteur

462 L. Casian, R.J. Stanton

is at most p. Then {Xp} is a filtration of X which, subject to suitablevanishing conditions, leads to a chain complex of the form

Cell∗({Xp};Z) = · · ·→Hp(Xp, Xp−1;Z)→Hp−1(Xp−1, Xp−2;Z)→· · ·whose homology isH∗(X;Z). In Section 4 we formulate this constructionin sheaf-theoretic terms and we relate it to cellular resolutions.

The basic problem is to obtain suitably explicit information on the dif-ferentiald in Cell∗({Xp};Z) to compute the homology. The Schubert cellsare parametrized by a subsetWP of the Weyl group. To computed first weintroduce in (7.4) a relation⇒ between pairs of elements inWP. This rela-tion, essentially combinatorially defined, refines the Bruhat order inWP andreflects the influence of theR-form G on the usual Bruhat order. Topologi-cally, the relation⇒ contains information concerning the non-orientabilityof certain submanifolds ofG/P, and for complex groups is not present. Thisrelation leads in (7.5) to an oriented graph,G∗, and, when supplementedwith a collection of signs, to a complex.

To connect with representation theory, we associate aG-module to eachcomplex Schubert cell. The collection of these will constitute aG-complexquasi-isomorphic to the De Rham complex ofX = G/P, thus suitablefor rational (co-)homology. Since the complex consists ofG-modules, thegeometric formulation of representation theory associates to this complexof G-modules a complex of perverse sheaves on the flag manifoldGGG/BBB.Here we can compute the differential as we implement the Vogan calculusin a sheaf-theoretic formulation. Crucial to the analysis here, besides thework of Vogan, is that of Lusztig–Vogan ([LV]), and various results ofBernstein. Finally, in Section 7 we show that the preceding complex ofperverse sheaves can be constructed with general coefficients resulting ina complexC

•Z from which one can compute the integral (co-)homology.

This, coupled with the results in Section 5, allows us to make the transitionfrom the rational (co-)homology to the integral (co-)homology ofX.

Our major results involve a formulation in representation theory of topo-logical properties ofX. From Cartan, one knows that the Betti numbers arerelated to the occurrence in the De Rham complex of the trivial representa-tion of the maximal compact subgroupK . First we relate the Betti numbersto the occurrence as a subquotient of the trivial representation ofG. Then asa feature unique to the real formG, we introduce a substitute for the trivialrepresentation – that ofasymptotically largerepresentations. Theasymp-totically largerepresentations will provide the connection of representationtheory ofG to torsion in the cohomology, and the representation-theoreticsetting for most of our topological results.

For example, we introduce in (5.10) a second relation inWP denoted→→. Roughly speakingx →→ y if an intertwining map involving twoinduced representations attached tox and y has an asymptotically largerepresentation in its image. By means of this relation, the Schubert cellsalso give rise to an oriented graph denotedGrep∗ . The relationship of the

Schubert cells and representation theory 463

two graphs,Grep∗ andG∗ is clarified by comparing (5.11) and (7.3) to see

that as oriented graphs they are equal. The proof of this uses a notion ofgroups withample involutionsdescribed in § 2 where it is shown that if theLie algebrag0 of G does not contain a summand of specific types, thenG hasample involutions. Those simple Lie algebras not admittingampleinvolutionsare handled in (2.11) by an argument involving Weyl groups.As a consequence we obtain in (7.11)

Theorem A. The Schubert cellc in G/P, represented byw ∈ WP, isa cycle if and only ifw has minimal length in its connected component inthe graphG∗. Moreover, a Schubert cellc determines a non-zero integralhomology class[c] if and only ifc is a cycle.

For another re-formulation of topological information, we let{πδ} bethe set of principal series modules in standard position and with trivial in-finitesimal character, and letπ(δ) be the unique irreducible submodule ofπδ. The parameterδ will denote, for us, a Beilinson-Bernstein parameteras in their ([BB]) version of the Langlands classification. Thusδ consistsof a KKK orbit in the flag manifoldXXX together with a certain local systemon it. If {J(ξ(w)) : w ∈ WP} is the collection ofG-modules associated tothe Schubert cells, we introduce in § 7 the notion of a compatible fam-ily of intertwining maps,R(·), of J(·) into the standard principal seriesrepresentations{πδ}. These can be made into a complex of intertwiningoperators (a consequence of (7.8)) with differential defined using⇒ from(7.4), and whose cohomology is denotedHk(G∗;Z,e), in the notation in-troduced in § 7. We can state our main result that computes the integral(co-)homology ofG/P with a representation-theoretic complex.

Theorem B. H∗(G/P;Z) and the cohomology of the dual complex ofintertwining operators4nt are isomorphic.

This result is proved in two steps. First, as a consequence of (7.8) weshow there is a surjectionHk(G∗;Z,e) � Hk(G/P;Z) whose kernelconsists of torsion classesc such that for some odd integern, nc= 0. ThusTheorem B follows if in the cohomology of the complex of intertwiningoperators we show there are no torsion elements killed by powers of oddprimes. This we finally do in (9.9) as an application of the technique ofreduction to characteristicp.

Basic to the above is our computation of the differential in the complexof intertwining operators and our results concerning groups with ampleinvolutions. Our description of the differential is summarized in

Theorem C. The cohomology ofG/P with coefficients inZ is the cohomo-logy of a complex

· · · →⊕w∈WP

l(w)=k

Zw→⊕w∈WP

l(w)=k+1

Zw→ · · · .


The differentials are matrices whose entriesa(x, y), l(x) = k, l(y) = k+ 1,are in {0,±2} and satisfy:

a(x, y) = 0 ⇔ x 6⇒ y

anda(x, y) = 2s(ξ(y), ξ(x)) if x⇒ y.

In particular, one concludes from Theorem C that each non-zero entrya(x, y) in the matrix of these differentials is always related through definition(7.4) to a specific real root for a maximally split CartanH, and whose asso-ciated homomorphismΦα : SL(2, R)→ G is such thatΦα(Diag(−1,−1))is not in the connected component of the identity of a maximally split Car-tan H. It follows that if G does not have such real roots, for instance ifG = SU(2,1), thenG/P cannot have torsion for anyP. For suchG thecohomology of anyG/P can be described very easily, since all the differ-entials in the chain complex in Theorem C are zero. In this way we are ableto handle, by means of (2.11), real simple Lie groups that fail to have ampleinvolutions. For such groups one has

Corollary of C. Let G be such that for every real rootα with respect toa maximally split CartanH, Φα(Diag(−1,−1)) is contained in the identitycomponent ofH. Let P be a parabolic subgroup ofG. ThenHk(G/P;Z) ∼=Zs(k), wheres(k) is the number of elements inWP of lengthk.

Returning to the original motivation of computing the integral homologystarting from the Schubert–Bruhat decomposition, it becomes necessarythen to relate the complex of perverse sheaves in a direct way to the complexobtained from the Schubert cell filtration. The complex flag manifoldGGG/BBBis the base of a fibration with typical fiberXC = KKK/KKK ∩ PPP, andGGG/BBBhas a natural filtration by unions ofPPP-orbits. The pull-back toXC of thisfiltration ofGGG/BBB gives a filtration,{Yp}, of XC with a resulting cell complex,Cell∗({Yp};Z). We have the cellular chain complex Cell∗({Xp};Z) for X,and the chain complex Cell∗({Yp};Z) for the complex manifoldXC . Somework (§ 4) is needed to show thatXp = Yp ∩ [K/K ∩ P], and to determine(§ 5) the relationship between the two sheaf complexes, resulting in (9.10) in

Theorem D. There is a chain isomorphism between the chain complexassociated to a compatible family of intertwining operators

· · · →⊕w∈WP

l(w)=k

Zw→⊕w∈WP

l(w)=k+1

Zw→ · · ·

and the cellular chain complex

Cell({Yp},Z) free= · · · → H p(Yp,Yp−1;Z) free

→ H p+1(Yp+1,Yp;Z) free→ · · · .


Restriction maps induce a quasi-isomorphismCell({Yp};Z) →Cell({Xp};Z).ExampleSL(3,R)/P, P a minimal parabolic subgroup.While this ex-ample is a bit misleading because of its simplicity, nevertheless, it serves toillustrate some of the preceding results. Note that herea⇒ b is equivalentto the non-orientability of the union of the corresponding Schubert cells;however, this interpretation of⇒ fails already in theSL(4,R) examplein (8.1). The Weyl group consists of{ e,1,2,12,21,121} with 1, 2 simplereflections. We have only 1⇒ 21, 2⇒ 12. For the graphG we denoteby a box the elements that contribute to rational cohomology. The corres-ponding representations,J(e) and J(wmax) are the only ones that containthe trivial representation ofG as a subquotient.

e1 2⇓ ⇓21 12wmax

From the dual graphG∗ and Theorem A, one obtains that the Schubertcells that are not cycles are those corresponding to 12 and 21. In fact theclosures of these two cells are copies of the Klein bottle.

Applying Theorem C we obtain the integral cohomology groups:

i) H0(G/P;Z) = Zii) H1(G/P;Z) = 0iii) H2(G/P;Z) = Z/2Z⊕ Z/2Ziv) H3(G/P;Z) = Z.

Acknowledgements.We want to express our sincere gratitude to the referee whose manyquestions and comments on earlier versions of this paper prompted us to clarify and tocorrect several points.

1. De Rham complex

In this section we shall introduce the rather standard notation that will beused and we shall recall various facts about generalized Verma modules,leading up to a complex of principal series representations that refines theDe Rham complex.

Notation.

Let GGG denote a complex, connected, reductive, linear group definedoverR, and letG denote the identity component of the real points ofGGG.Choose a maximal compact subgroupK of G and letθ be a Cartan involution


on G with K = Gθ . Let H = TA be aθ-stable maximally split Cartansubgroup for whichT = Hθ andθ acts by−1 on the Lie algebra ofA. Let Pbe a parabolic subgroup containingH, and denote the Levi decompositionby P = LU, L a θ-stable Levi factor andU the unipotent radical. Thegeneralized flag manifoldG/P ' K/K ∩ P will be denoted byX and itsdimension byn.

We shall denote the Lie algebra ofG by g0, and its complexificationby g. Similarly, h is the complexification of the Lie algebra ofH, p of P,etc. In p0 we may choose a minimal parabolic subalgebra ofg0 and letpmin be its complexification. Then letb = h ⊕ n be a Borel subalgebracontained inpmin, i.e.p ⊇ pmin ⊇ b. Let n+ be the nilradical ofpmin, withN+ the corresponding subgroup ofG, and letb∗ = h⊕ n∗ denote the Borelsubalgebra opposite tob. Now n determines a system of positive roots∆+for (g, h), and∆+L for the Levi factor(l, h). As is customary, letρ be theweight given by one-half the sum of the roots in∆+.

Weyl groups.

Denote byWGGG (resp.WPPP) the Weyl group of∆+ (resp.∆+L ), and bySthe set of simple reflections determined by∆+. The Weyl groupWGGG hasa longest element saywg, and, similarly, forWPPP there is awl. Set

WPPP = { x ∈ WGGG

∣∣x∆+ ⊇ ∆+L}.

ThenWPPP consists precisely of thosew in WGGG for whichρw = wρ− ρ is∆+Ldominant, or, equivalently,WPPP is a cross-section toWGGG → WPPP\WGGG withimage minimal length representatives. Similarly we can take a cross-sectionto WGGG → WGGG/WPPP with image minimal length representatives. Denote thisset byWPPP. Then the two setsWPPP andWPPP can be identified under a bijectionξ : WPPP→ WPPP, σ → wlσ

−1wg, with inversew→ wgw−1wl. Then

WPPP = { x ∈ WGGG

∣∣wlx−1wg ∈ WPPP

}.

We note some elementary relationships ofWPPP and WPPP. For example,e ∈ WPPP corresponds towgwl, whilewlwg ∈ WPPP corresponds toe ∈ WPPP.We let σmax = wgwl denote the longest element inWPPP. The rationale forintroducing these two sets of coset representatives is thatWPPP will natu-rally parametrize certain representations (i.e. cohomology), whileWPPP willnaturally parametrize certain cells (i.e. homology).

The use of a subscript such asG on a Weyl group (or a subset of one)will mean that representatives are fromG. Thus one hasWG the Weylgroup of (g0, h0) a subgroup ofWGGG, and WP the Weyl group of(l0, h0)

a subgroup ofWPPP. Similarly let WP be those elements inWPPP that haverepresentatives inK . The setWP parametrizes the real Schubert cells in the


Bruhat decomposition

G =⋃w∈WP

N+wP

where the cell associated tow has dimensionl(w). We setWP = ξ(WP).We caution that in generalWP does not consist of those elements inWPPP

that have representatives inK (equivalently inG). Occasionally all thesesets, introduced above, are equal, for example, ifG is split overR andP isa minimal parabolic subgroup, thenWPPP = WGGG = WG = WPPP

G . However ingeneral, all these sets are different.

H-characters.

For anyw ∈ WPPP, sayw = ξ(w), the vector spacenw = wn ∩ n∗,is the complexified cotangent space of the Poincare dual to the complexSchubert cell associated tow. The disconnected groupH acts on the topexterior power∧l(w)nw ; moreover,T often acts non-trivially. This actionhashweightρw = wρ− ρ. We denote byχ(w) the character ofH given bythis action on∧l(w)nw. For example, forσmax = wgwl, the longest elementin WPPP, the characterχ(e) = χ(ξ(σmax)) is trivial on T. Indeed forσmax,w = e, and the module isC.

The disconnectness ofH introduces some important technicalities. AsH is maximally split and abelian (G is linear), there is ([Wa] 1.4.1.3) a finitesubgroup ofH, sayF†, so thatH ∼= Ho F†. Also, ([Wa] p.74) one has thatK∩P = K∩L = (K∩L)o F†, soK∩P/(K∩P)o ∼= F†/(F†∩(K∩L)o) ∼=H/H for someHo < H < H.

For any character ofK∩L/(K∩L)o ∼= F†/(F†∩(K∩L)o) ∼= H/H weobtain a locally constant sheaf (a local system) onK/K ∩ L = G/P. Oneof these charactersζo, theorientationcharacter, gives rise to the orientationsheaf involved in Poincare duality. We will not need the explicit descriptionof this character.

BGG order.

In [BGG] a partial order onWGGG was introduced together with a consistentchoice of signs. One of the results of this paper will be an improvement ofthis partial order that incorporates data from the real formG. We follow thenotation of [BGG] or [Le].

(1.1) Definition. Letw1 andw2 be elements ofWGGG. We writew1 → w2 ifthere is a reflectionsα such that

w1 = sαw2,(i)

l(w1) = l(w2) + 1.(ii)

More generally, we writew < w′ if there arew1, . . . , wr with w → w1,wi → wi+1 andwr → w′.


(1.2) Definition.We call a quadruple of elements(w1, w2, w3, w4) a squareif we have

w1↙↘

w2 w3↘↙w4

hencel(w4)+ 1= l(w2) = l(w3) = l(w1)− 1.

(1.3) Definition. Let s: WGGG x WGGG → {±1,0} be the function defined in[BGG] Lemma 10.4 with the property thats(w,w′) = 0 if w ≮ w′, andmoreover for every square the product of the signs for the four arrows isequal to−1.

The sign function need not be unique. In fact, while we use the existenceof one choice of it now, later in § 7 we will show that one can revise thechoice to reflect structure associated to the real parabolic.

Generalized Verma modules.

Let V be a highest weight module forp = l⊕u with trivial u-action anda compatibleL ∩ K action. Set

M(V) = U(g) ⊗U(p)

V .

This generalized Verma module is ag-module with a compatible action ofL ∩ K .

For σ in WPPP, let V[χ(ξ(σ))] be the irreducible, finite-dimensionalLmodule with the highest weightρξ(σ) and H action on the highest weightspace according to the characterχ(ξ(σ)), (due to the disconnectednessof H one needs to keep track of not just the highest weight but rather theH-action on the highest weight space). Then for eachσ in WPPP (equivalentlyw ∈ WPPP) one has the generalized Verma moduleM(V[χ(ξ(σ))]) that invarious contexts we abbreviate asM(χ(ξ(σ)) or M(w), w = ξ(σ) ∈ WPPP

where theH−character on∧l(w)nw is χ(ξ(σ)).Maps between generalized Verma modules are (g, L∩K )-module maps.

Amongst these are those referred to in [Le] as standard maps. From [Le]we recall then that a standard map is a(g, L ∩ K )-module map betweengeneralized Verma modules,

f(w,w′) : M(w)→ M(w′),

herew andw′ are inWPPP, obtained from a map between the correspondingVerma modules of highest weightsρw andρw′ respectively by passing toa quotient. We will considerw andw′ having lengths that differ by one. Dueto the presence of the [BGG] sign function in the definition of the differentialwe need only be concerned then with a standard map forw andw′ in WPPP

withw→ w′. In this case it is non-zero (Proposition 3.7 in [Le]). These are


unique up to a scalar and all these scalars can be adjusted so that ifM(w)maps toM(w′) andM(w′) to M(w′′), then f(w,w′′) = f(w′, w′′)◦ f(w,w′).

BGG resolution.

The BGG resolution as extended by Lepowsky ([Le]) is a resolution ofthe trivial(g, L∩K )module by generalized Verma modules and(g, L∩K )-module maps. It has the form:

. . . →⊕σ∈WPPP

l(σ)=k

U(g) ⊗U(p)

V [χ (ξ(σ))](1.4)

→⊕τ∈WPPP

l(τ)=k+1

U(g) ⊗U(p)

V [χ(ξ(τ))] → · · · → C→ 0

Here, the mapM(w) → M(w′) is given by sendingv → s(ξ(σ), ξ(τ))f (ξ(σ), ξ(τ)) v with f(w,w′) the standard map betweenM(w) andM(w′)described above ands(ξ(σ), ξ(τ)) as in (1.3). We will refer to the collectionof maps of the formv → s(ξ(σ), ξ(τ)) f(ξ(σ), ξ(τ))v as a BGG familyof maps. Clearly such a collection depends on theBGG signs chosen. Inturn, the signs can be recovered from this collection of maps. In § 7 wewill consider a projective functorF as in [BG], such thatF(M) is a Vermamodule wheneverM is a Verma module. The functorF will be appliedto a BGG family of maps. The resulting collection of maps will also bereferred to as aBGG family of maps.

If we take duals and then theL ∩K -finite parts of the resulting modules,we obtain a resolution involving duals of generalized Verma modules:

0→ C→ . . .⊕σ∈WPPP

l(σ)=k+1

Homp(U(g),V[χ(ξ(σ))]c)L∩K(1.5)

→⊕τ∈WPPP

l(τ)=k

Homp(U(g),V[χ(ξ(τ))]c)L∩K → · · ·

The “c” as superscript denotes the contragredient representation.

Computation ofH∗(G/P;C) and I* (BGG).

We shall use the functor, introduced by Zuckerman,Γ from the categoryof compatible(g, L ∩ K )-modules to the category of(g, K)-modules (see[Vo]) as well as the functor, introduced by Enright–Wallach and Bernstein,L from the category of compatible(g, L ∩ K )-modules to the categoryof (g, K )-modules (see [Bi] p.39-40). The functorΓ is left exact while the


functorL is right exact, and they satisfy the following adjointness condition:let M be a compatible(g, L ∩ K )-module, then

Γ(Mc) = (L M)c.

Herecdenotes the natural dual object in the appropriate category, namelyMc

is theL∩K locally-finite part of HomC (M,C), and(L M)c is the dual objectin the category of Harish–Chandra modules. In particular,L M = Γ(Mc)c

if L M is admissible.

(1.6) Proposition. Let X be a compatible(g, L ∩ K)-module of the formHomp(U(g),V[χ(w)]c)L∩K . Then the derived functorsRqΓ(X) = 0,q> 0.

Proof.Γ can be described by

ΓX =⊕

Z irreducibleK−module

Hom(k,L∩K )(Z,Res(X))⊗ Z,

where Res(X) denotes restriction ofX to a(k, L ∩ K )-module. Then, as in[Vo] (Prop. 6.3.5) one observes that

Homk,L∩K (Z,Homk∩l(U(k),V[χ(w)]c)L∩K) = HomL∩K(Z,V[χ(w)]c) .From this it follows that Homp(U(g),V[χ(w)]c)L∩K is injective asa (k, L ∩ K )-module. Hence the derived functorsRqΓ(X) = 0. ut

The following result is well known, but we include it because of itsrelevance to our presentation.

(1.7) Proposition. Let F denote the trivial(g, L ∩ K )-module. Then

RqΓ(F) = Hq(K/L ∩ K;C).

Proof. Take any injective resolutionJ∗ of F in the category of compatible(k, L ∩ K )-modules. Then for each irreducibleZ, the complex Hom(Z, J∗)computes Ext groups betweenZ andF. Of course, these all are zero unlessZ = F. Therefore in the category of (k, L ∩ K )-modules,RqΓ(F) =Extq(F, F), so we obtain relative Lie algebra cohomology groupsHq(k, L∩K; F) which are known from Cartan to agree withHq(K/L ∩ K;C). ut

Since theBGGresolution, in terms of duals of generalized Verma mod-ules, is a resolution ofF, the trivial representation, and we have that all thehigher derived functorsRqΓ of the modules Homp(U(g),V[χ(w)]c)L∩K arezero, we obtain as a consequence


(1.8) Proposition. There is a complex of (g, K )-modules, of the form

· · · →⊕σ∈WPPP

l(σ)=k+1

ΓHomp

(U(g),V [χ(ξ(σ))]c

)L∩K

→⊕τ∈WPPP

l(τ)=k

ΓHomp

(U(g),V [χ(ξ(τ))]c

)L∩K→ · · ·

whose cohomology computesH∗(K/L ∩ K;C) = H∗(G/P;C).This result, apparently, was known to Zuckerman and it may be found

in the thesis [RC]. We thank N. Wallach for informing us of this reference.Denote byI(w) the(g, K )-moduleΓHomp(U(g),V[χ(w)]c)L∩K . Then

I(w) can be identified with a principal series representation, induced fromthe parabolicP.

Thus we have obtained a complex of(g, K )-modules and(g, K )-modulemaps

0→ C→ · · · →⊕σ∈WPPP

l(σ)=k+1

I(ξ(σ))→⊕τ∈WPPP

l(τ)=k

I(ξ(τ)) → · · ·(1.9)

(1.10) Definition.We denote the complex of principal series representations(1.9) byI ∗(BGG).

Computation ofH∗(G/P;C) and J* (BGG).

Dually, one can proceed from generalized Verma modules and showthat they are projective as(k, L ∩ K )-modules. If one would apply thefunctor L to theBGG resolution by generalized Verma modules, then onewould obtain a complex of(g, K )-modules involving the modulesL(M(w)),w in WPPP. This complex will be dual toI ∗(BGG) because of the relation(L M)c = Γ(Mc).

If we tensor the BGG resolution in (1.4) with a characterζ of L ∩ K/(L∩K )o, we obtain a resolution of a compatible one dimensional(g,L∩K )-moduleCζ whereL∩K/(L∩K )o ∼= H/H may act with a non-trivial action.As the characterζo defines theorientation sheaf ofG/P, it is possibleto see from (1.8) that this character determines orientability. Indeed theorientability ofG/P depends on the presence of the trivialK representationin the last term in (1.8), which by Frobenius reciprocity is determined byζo.

(1.11) Definition.With a slight abuse of notation we denote byJ(w), w =ξ(σ), the moduleL(U(g)) ⊗

U(p)V[χ(ξ(σ))⊗ ζo]. The chain complex obtained

by first tensoring (1.4) withζo and then applyingL is denoted byJ∗(BGG).


The chain complexJ∗(BGG)[−n] also computes the rational cohomo-logy of G/P. This will follow from (3.7) in § 3 as a consequence of the factthat

J∗(BGG)[−n] ∼= I ∗(BGG),

in particular, J(w) ∼= I(wlwwg). Alternatively, this follows from a ver-sion of (1.8) with local coefficients (the orientation sheaf). We start with(1.8) with twisted local coefficients, the orientation sheafCζo. Then(Hk(G/P;Cζo))∗ = H−k(J∗(BGG)). From Poincare duality,(

Hk(G/P;Cζo))∗ ∼= Hn−k(G/P;C).

So J∗(BGG)[−n] computes the cohomology ofI ∗(BGG), namelyH∗(G/P;C).

2. Relative position

We begin an examination of the structure of the differentials inI ∗(BGG)and J∗(BGG) by presenting two constructions of standard maps betweengeneralized Verma modules that subsequently we show can be realizedgeometrically.

Translation functors.

Let g(ρ) denote the category ofg-modules with trivial infinitesimalcharacter. In particular the modulesM(w), I(w) andJ(w) are ing(ρ). Letαbe a simple root in∆+. If β is a root in∆+, let Hβ be the element in[gβ, g−β]with β(Hβ) = 2. Define a weightωα byωα(Hβ) = δαβ for any simple rootβ.Following Jantzen and Zuckerman we consider the two translation functors

ψα : g(ρ)→ g(ρ − ωα)φα : g(ρ − ωα)→ g(ρ) .

The functorψα is defined in two steps. First one forms the tensor prod-uct with a finite dimensional irreducible representation ofg of lowestweight−ωα. Then one projects to the direct summand with generalizedinfinitesimal characterρ − ωα.

For the functorφα, similarly there are two steps. First tensor with a finitedimensional representation of highest weightωα and then project to thesummand with generalized infinitesimal characterρ.

These two functors are exact and satisfy an adjunction relation describedby the following isomorphisms

HomU(g)(ψαX, ψαY) ∼= HomU(g)(X, φαψαY) ∼= HomU(g)(φαψαX,Y).(2.1)

Also, the relationship of the translation functors and the functorΓ is givenin [Vo] (p. 489).


Let s ∈ S be a simple reflection that corresponds to the simple rootα.Denote byψs the functorψ−wgα and byφs the functorφ−wgα. Note thatfor any moduleX in g(ρ) with dual objectX∗ in g(ρ) we have(ψsX)∗ ∼=ψα(X∗). Although frequentlyψs = ψα, the constructionφsψs, rather thanφαψα, is more relevant to the fibrationπs of § 5.

(2.2) Definition. We say that the pair ofg-modules(X,Y) from g(ρ) is inrelative positions, if

a) ψsX ∼= ψsY

b) φsψsX→ X is surjective

c) Y = Ker(φsψsX→ X).

Thus there is a short exact sequence

0→ Y→ φsψsX→ X→ 0.(2.3)

(2.4) Lemma. Letw andw′ be inWPPP withw′ → w. Suppose thatw′ = sw,for somes ∈ S. Then the pair ofg-modules(M(ξ(w)),M(ξ(sw))) is inrelative positions. Hence there is a short exact sequence:

0→ M(ξ(sw))→ φsψsM(ξ(w))→ M(ξ(w))→ 0 .

Proof. This is a well-known fact for which we have no specific reference.We sketch the proof noting that the necessary details can be gleaned, forexample, from [VoII]. From the isomorphism

U(g) ⊗U(b)

(L ⊗ Cλ) ∼= L ⊗U(g) ⊗U(b)

Cλ,

where L is a finite dimensionalg module, one determines the allowableweights that can occur, and thus thatψ−wgαM(ξ(w)) ∼= ψ−wgαM(ξ(sw)).To consider the functorφ−wgα, take L with lowest weight−ωα (highestweightω−wgα ). To simplify our sketch take the case whenw = e, p = b.Observe thatL contains the weights−ωα, −sαωα. Now these two weightsform a two dimensionalb quotient which gives rise to the short exactsequence, with each weight producing a Verma module. ut(2.5) Corollary. With s andw as above, the pairs ofg-modules(J(ξ(w)),J(ξ(sw))), and (I(ξ(sw)), I(ξ(w))) are in relative positions. Hence thereare short exact sequences

0→ J(ξ(sw))→ φsψsJ(ξ(w))→ J(ξ(w))→ 0

and0→ I(ξ(w))→ φsψsI(ξ(w))→ I(ξ(sw))→ 0 .


Proof. This follows from the left-exactness of the functorΓ, the vanishingof the higher derived functorsRqΓ, and Lemma (2.4). ut

This notion can be extended to two pairs ofg-modules. Let(X,Y)and (X′,Y′) be two pairs ofg-modules with trivial infinitesimal charac-ter. Suppose that each pair ofg-modules is in relative positions. Thenag-map f : X → X′ induces a mapf(s) : Y → Y′. Indeed, f gives riseto φsψs f : φsψsX→ φsψsX′ and there is a commutative diagram resultingfrom relative positions

0−→ Y −→ φsψsX −→ X −→ 0yφsψs f

y f

0−→ Y′ −→ φsψsX′ −→ X′ −→ 0

Standard diagram analysis produces a mapf(s) from Y to Y′.We extend these considerations fromw′ → w to elementsw andw′ with

w < w′. One should probably extend the notion of “relative positions” fora pair ofg-modules to one of “relative positionw” for an arbitrary elementin the Weyl groupWGGG. However, this would mean having to check thatcertain constructions are independent of the choice of a reduced expressionof a Weyl group element. For our purposes, it suffices to fix a reducedexpressionw = s1 . . . sn and define “relative positionw”.

(2.6) Definition. We say that the pair ofg-modules(X,Y) each ing(ρ) isin relative positionw if there is a sequenceXn, Xn−1, . . . , X0 of g-modulesfrom g(ρ) such thatX = Xn, (Xi , Xi−1) are in relative positionsi for all i ,and X0

∼= Y.

Similarly this can be extended to pairs ofg-modules fromg(ρ). Let(X,Y) and(X′,Y′) be two pairs ofg-modules each in relative positionw.As before, given ag-module mapf : X→ X′ one obtains ag-module mapf(w) : Y→ Y′ defined inductively byf(s1 . . . sn) = f(s2 . . . sn)(s1).

Remark.For I ∗(BGG) it suffices to understand standard maps forw, w′ inWPPP, w′ → w (or dually for J∗(BGG), w → w′). In this casew′ = swwith either s ∈ S or elses is a reflection not corresponding to a simpleroot. In either case, it follows thatw � w′ in the Bruhat order. Fors /∈ Sthere is a reduced expression ofw′ in terms of reflections coming fromelements ofS,w′ = s1 · · · sm such thatw = s1 · · · sj · · · sm . Heresj meansthatsj has been deleted from the expression. Now takex = s1 · · · sj−1 andy = sj+1 · · · sm. We obtainw = xy, w′ = xsj y. From Proposition 3.5 of[Le] concerning “initial segments” inWPPP combined with the observationthat the mapξ : WPPP→ WPPP reverses initial segments, it follows that bothyandsj y are inWPPP. Moreoverx increases the length of bothy andsj y byexactly the length ofx.

(2.7) Lemma. Letw andw′ be inWPPP withw′ → w. Suppose thatw = xy,w′ = xsy, for somes ∈ S. Letx be a reduced expression ofx. Given a stand-


ard map f : M(ξ(y)) → M(ξ(sy)), then f(x) : M(ξ(w)) → M(ξ(w′)) isa standard map. In particularf(x), in this case, depends (up to a non-zeroscalar) only onx.

Proof. The proof follows easily from the previous remark and Proposition3.7 in [Le]. utInvolutions associated to real roots.

We suppose for the moment thatG is a split group and thatg issimple. Let{αi } be a set of simple roots and let{Hωi } be a dual basis,i.e. α j (Hωi ) = δ

ji . As all roots are real here, theHωi are ina0. For each

αi consider the automorphism ofg given by AdeπiHωi wherezαi = eπiHωi

is in the maximal compact subgroup ofGGG. AdeπiHωi on h is the iden-tity; Ad eπiHωi (Xαi ) = eπiαi (Hωi )Xαi = −Xαi ; while, for any other rootδ,Ad eπiHωi (Xδ) = eπiδ(Hωi )Xδ = (−1)mi Xδ wheremi is theαi coefficient ofδ.Let τ denote conjugation ofg relative tog0. Then as eachHωi is in a0,

Ad eπiHωi ◦ τ = τ ◦ Ad eτπiHωi = τ ◦ Ad e−πiHωi .

However for any rootδ, δ(2πiHωi ) ∈ 2πiZ, thuse2πiHωi is in the center ofthe maximal compact subgroup ofGGG and so acts trivially ong. Thus wehave

Ad eπiHωi ◦ τ = τ ◦ Ad eπiHωi ,

and so AdeπiHωi preservesg0. The same argument with the Cartan involu-tion θ shows that AdeπiHωi preservesK . As the parabolic subalgebrap0 isstandard, and on root spaces AdeπiHωi = ±1, i zαi

, conjugation byeπiHωi ,preservesK ∩ L. Thus to the simple rootαi there is an involutionσαi of gimplemented byzαi = eπiHωi and with properties:

σαi |h = identity,σαi (X±αi ) = −X±αi ,

σαi (Xδ) = ±Xδ,σαi (k0 ∩ l0) = k0 ∩ l0,

σαi preserves the nilradical of any standard parabolic subalgebra.

The restriction to simple roots is easily removed. Given any rootα thereis an element,w, of the Weyl group and a simple root,αk, withw−1α = αk.As {wHωi } is a dual basis to{wαi }, the previous argument also shows thereis an involutionσα of g implemented byzα = eπiwHωk having the propertieslisted above withα in place ofαi .

Clearly, the restriction to simple Lie algebras was unnecessary and wemay assume thatg is semisimple. Thus in the case of a split groupG andfor any (real) rootα there is an involution ong that we may denoteσα,implemented by an elementz, with the following properties

σαX±α = −X±α,


while onh, σα acts by the identity. We have thatσα preserves the nilradicalof any standard parabolic subalgebra andL ∩ K (resp. K ) is preservedunderi z, conjugation byz.

The action of the involutionσα can be extended to modules. IfX isa g-module then we can obtain anotherg-module structure onX by pre-composing with the automorphismσα. We shall denote byXσα the newg-module obtained in this way. IfX has a compatible action ofL∩K (resp.K )then we will use the elementz introduced above and the inner automorphismi z to twist the corresponding group action. Thusσα, consisting of the pair(Ad z, i z), gives a morphism of(g, L ∩ K ) (resp.(g, K ))-modules. In par-ticular for SL(2,R) if X is a module forsl(2,C), then we can twist withσto obtain another moduleXσ . Also, if X is an(sl(2,C), SO(2))-module orcategoryO-module forsl(2,C), then so is the twisted moduleXσ .

(2.8) Lemma. Assume thatG is a split group andh is a split Cartansubalgebra ofg0. Letα be a (real) root forh. Forw ∈ WPPP, let M(w) be thecorresponding Verma module andM(w)σα the(g, L ∩ K )-module obtainedby twisting by the involutionσα. Then there is a(g, L ∩ K )-isomorphism

T : M(w)σα ∼= M(w).

Proof.Notice that the finite dimensional moduleV[χ(w)] is a simple modulefor PPP. We claim thatV[χ(w)]σα is isomorphic toV[χ(w)] asPPP-modules.This is a consequence of the highest weight theory applied to the complexconnected groupLLL, the Levi factor ofPPP. As the automorphismσα is theidentity onh, the two representationsV[χ(w)]σα andV[χ(w)] have the sameweight spaces and the same H-action, hence the same character. There is thena unique (up to scalar) mapT that realizes this isomorphism. This mapTintertwines the twoPPP actions and in particular it gives a(p, L ∩ K )-modulemap. Now a map from the generalized Verma moduleM(w) to M(w)σα isdetermined by a mapV[χ(w)] → M(w)σα viewed as(p, L ∩ K )-modules.Thus the desired map is the identity betweenV[χ(w)] and V[χ(w)] nowviewed as part ofM(w)σα . This map produces the isomorphism. ut(2.9) Lemma. Assume thatG is a split group andh is a split Cartansubalgebra ofg0. Letα be a (real) root forh. Letw ∈ WPPP. Then the mapTin (2.8) induces an isomorphism of(g, K ) modules

I(w)σα ∼= I(w).

Proof.By (2.8) the two(g, L∩K )-modulesM(w)σα andM(w)are equivalentby the mapT. Therefore, their linear duals are also equivalent. If we applyΓ to both sides we get the desired isomorphism. Notice that it is essentialthatσα = (Ad z, i z) preserveL∩K andK , and thatΓ is given by(k, L∩K )maps. ut


Remark.Before proceeding to the general case we note that (2.8) and (2.9)are valid for anyG and any real rootα for which there is an involution withthe properties ofσα.

We return now to the general case as described in § 1 so thatGGG denotesa complex, connected, reductive, linear group defined overR, andG is theidentity component of the real points ofGGG. We remark that these conditionsare consistent with those in [K-Z]. We recall thatH is a maximally splitθ-stable Cartan subgroup ofG and thath is the complexification of the Liealgebra ofH. Let α be a real root of(g, h). Motivated by the discussion inthe split case we make the following definition.

Definition. Given a real rootα for h, an involutionσ of (g,G, h) is anample involution forα if σ(X±α) = −X±α, σ acts as the identity onh,σ preservesK and for any parabolic subgroupP, containingH, with Levifactor L, σ preservesL ∩ K andu. A groupG such that each real rootαfor h has an ample involution forα will be referred to as being a group withample involutions.

Our goal is to show for simpleG thatG either has ample involutions orG has a real root relative tohwith one-half this root a restricted root. Giventhe earlier discussion, we may and will suppose thatG is not split, i.e. inthe notation of [Wa] p. 30-32,G is not one of AI, BI (l+ = l), CI, DI (split),EI, EV, EVIII, FI, or G.

Associated toα is the partial Cayley transformdα : h = t⊕ a→ h∗ =t∗ ⊕ a∗ (we follow the discussion in [K-Z]). Letm∗0 be the Levi factor ofthe parabolic subalgebra ing0 corresponding toh∗. Thenm∗0 is cuspidal andα = dα(α) is a noncompact root of(m∗, t∗). Let pα, in the Weyl groupWMMM∗ ,be the associated reflection. Then two cases arise according to whether ornot pα is in WM∗ .

We consider first the case thatpα is in WM∗ . Sincem∗0 is cuspidal one cantake a representative ofpα from K ∩ M∗; however, a more refined choicecan be made. From Lemma 2.2 in [K-Z],pα is in WM∗ precisely whenpαcan be represented by Adz, with z an element of a specific form in thecenter ofMmin. Moreover from Lemma 2.1b in [K-Z] one sees thatz maybe taken fromF(Tmin) as in the split case and is in the group generated bythe γβ ’s, γβ = eπi β . Thus in this case too Adz defines an involution ongwhich we denote byσα. From [K-Z] (p. 404-405) this involution has thefollowing properties

σαXα = −Xα,σαX−α = −X−α,

while onh, σα acts by the identity. Forβ a real root, Adγβ preserves thenilradical of standard parabolics. Sincez is in the group generated by theγβ ’s it follows thatp0 is preserved by the involutionσα. Moreover as beforeL ∩ K (resp.K ) is preserved underi z, conjugation byz.


Given the real rootα for h, there is a Lie homomorphismΦα : SL(2,R)→ G and an isomorphismΦα(sl(2,C)) ⊂ g. On Φα(sl(2,C)) the twopossible involutions, namely the restriction of theσα from G and that con-structed previously usingzagree. Suppose moreover thats = sα is in S. Thenbandsbgenerate a parabolic subalgebrap(s) = l(s)⊕u(s)with [l(s), l(s)] =Φα(sl(2,C)) ∼= sl(2,C); moreover, sinceα is real,l(s) is the complexifi-cation of l(s)0 ⊂ g0 with [l(s)0, l(s)0] = Φα(sl(2,R)) ∼= sl(2,R). On hthe involutionσα is the identity and on[l(s), l(s)] = Φα(sl(2,C)) the invo-lutions agree, soσα gives an involution extendingAdz to l(s). Notice thatwhen referring to the involution on ansl(2) or its extension tol(s) usuallywe will omit the subscriptα. Sincel(s) ⊃ h, u(s) is spanned by root spacesfor h it follows thatp(s) is preserved. More generally, ifX is ag-module,then the functors of twisting withσα then restriction to[l(s), l(s)], and re-striction to[l(s), l(s)] then twist withσ are equivalent. These observationswill be used in § 6.

Next we still considerα a real root forh but now the case thatpα is notin WM∗ . Unlike the split case not all real roots in all groups will have suchan involution. We are grateful to the referee whose questions prompted usto clarify this point.

To begin we recall a Galois-type action on roots introduced by Satake,which, for convenience, we use as described in [Wa] (p.21-34). Recall thatτ denotes conjugation ofg relative tog0 and let be conjugation ofCrelative toR. If α is a root then so isατ defined byατ(H) = α(τ(H)). Asbefore we consider first the case ofg simple. Fix aτ-order on the roots andlet {αi } be the simple roots arranged so that the firstl0 are real, the nextl1roots haveατi = αi +γ whereγ is nonzero and is supported ont, the nextl2roots haveατj = αi +γ wherei = j + l2, the nextl2 roots haveατj = αi +γwherei = j − l2, and the remaining simple roots are a basis for the rootsof t. Recall that the restrictionsαi of the(αi + ατi )/2, i = 1, ...l0 + l1 + l2give a basis for the restricted rootsΣ. First we consider real roots that aremultiples of these simple restricted roots.

If 0 ≤ i ≤ l0, thenαi is real,Hωi is in a0, and we proceed as in the splitcase.

Next we examineαi with l0 < i ≤ l1. We consider first the case that 2αiis not a restricted root, henceαi + ατi is not a root. We may then supposethat(αi + ατi )/2 is a root and thus a real root, for otherwise there is no realroot a multiple ofαi . Under these conditions forαi and 2αi one can showfrom [Wa] thatαi = αi ± γ/2 andατi = αi ∓ γ/2 and that the multiplicityof αi is odd. We claim thatHωi can be taken ina0. Indeed an examinationof [Wa] p.30-32 shows that under these conditions onαi , the simple roots{α j } of (g, h) are of two types: those withατj = α j + γ whereγ is zero,or, when nonzero is supported ont, and the imaginary simple roots. To getHωi , asa0 andt0 are orthogonal and the imaginary simple roots supportedon t, it suffices to take a dual basis to{αi } in a0. Then we can proceed as in


the split case usingHωi . The types of simpleg here are BI (l+ < l), BII, CII(l+ = l/2), DI (l+ ≤ l − 2), DIII (l+ = l/2), EVI, EVII, and EIX.

If 2αi is a restricted root, then one can show from [Wa] that(αi + ατi )/2is not a root. Thus for there to be a real root which is a multiple ofαi wemust have thatαi +ατi is a root. From the classification one sees that the Liealgebrag0 is of type CII (l+ ≤ l−1

2 ) or FII. For these we will not producean involution.

Next we examineαi with l1 < i ≤ l2 so thatατi = αi+l2 + γ . Weclaim that αi = (αi + ατi )/2 is not a root. Indeed if it were a root, thenαi = ∑mkαk with mk ≥ 0 andmi ≥ 1. Consequentlyαi = ∑mkαk and,as αi is a simple restricted root, we must havemi = 1, andmk = 0 forαk not imaginary andk 6= i . So αi = αi +∑mkβk with βk imaginary.Thenατi = αi = ατi −

∑mkβk = αi+l2 + γ −

∑mkβk and soαi+l2 = αi ,

putting us in the previous case. Thus we have that(αi + ατi )/2 is not a root,andαi + ατi is a real root, and 2αi is a restricted root. An examination ofthe classification shows thatg0 is of type AIII (Σ of type Bl), AIV, DIII(l+ = l−1

2 ), or EIII, and for these we will not produce ample involutions.We suppose then thatg0 does not have restricted rootsα and 2α where

2α is a real root. There are two possibilities for the simple roots. Eitherl2 = l1 in which case there are no simple roots withατj = αi + γ wherei 6= j . Then in this case for any real rootα there is an elementw of thesubgroupWτ of the Weyl group withw−1α equal toαi or r(αi +ατi ) and weproceed as in the split case to construct an ample involution forα. Hence forg0 not one of the above, and any real rootα there is an involution as in thesplit case. The other possibility isl2 6= l1. Those simpleg0 having simpleroots withατj = αi + γ wherei 6= j and the condition on the restrictedroots consist of quasi-splitg0, coming from types AIII (Σ of type Cl), DI,and EII. Here one can define an involution for each real root. Since theseg0are quasi-split, the Galois action isατj = αi . For simple real roots,i = j ,and as before we useHωi ∈ a0 to define an involution. Otherwise, for a realroot α, the easiest argument is to use the classification. Notice from ([Wa]p.30-32) thatατj is orthogonal toαi , and (from [Bour] Planches) that forαto be realα =∑miαi +∑ni (α j + ατj ) and wheremi is odd for some realrootαi . Then we useHωi ∈ a0 to define the involution.

Clearly the restriction to simpleg0 is unnecessary so that we may as-sume thatg0 is as in §1. The following is a consequence of the precedingdiscussion.

(2.10) Proposition. If G is a linear real reductive Lie group; andg0 doesnot contain any summand which has restricted rootsα and2α where2α isa real root, thenG is a group with ample involutions.

While g0 having restricted rootsα and 2α where 2α is a real root neednot have ample involutions, the Weyl groups of such groups have similarstructure that allows a uniform treatment of them. This is presented nextwith details in an appendix at the end of the paper.


The groupG(n) and the conditionG = G(a,b, c).

Let G(n) consist of all the permutationsx of (1, ...,n,−n, ...,−1) suchthat if x(i) = − j thenx(−i) = j . Each elementx of G(n) is determinedby a permutationσ of (1, ..,n) and a sign,ε(x) so thatx(i) = ε(x)(i)σ(i).

G(n) can be identified with the Weyl group ofSO(2n + 1) where thepositive roots areti − t j , ti + t j for i < j andti . Eachti represents a weightin the dual of a Cartan subalgebra of the Lie algebra ofSO(2n+1). Denotethis set of positive roots∆+(G(n)). The groupG(n) then acts on linearcombinations of theti by x(ti ) = ε(x)(i)tσ(i). Givenα ∈ ∆+(G(n)) there isa corresponding reflectionsα ∈ G(n).

The length of any elementx can be computed as follows. LetR1(x)be the number of roots of the formti − t j , i < j that become negativewhen x is applied; letR2(x) be the number of roots of the formti + t jfor i < j that become negative whenx is applied; and letR3(x) be thenumber of roots of the formti that become negative whenx is applied. Thelength ofx is thenL1(x) = R1(x)+ R2(x)+ R3(x). For anyy ∈ G(n) wedefine∆Ri (y, x) = Ri (yx)− Ri (x). We will fix positive integersa,b,c andintroduce a second “length ”L2(x) = aR1(x) + bR2(x) + cR3(x). We willassume however thata= b and eithera ≥ 2 andc≥ 2, ora= 0 andc≥ 2.Set∆Li (y, x) = Li (yx)− Li (x). The numbersa,b, c will be derived fromrestricted root multiplicities of certain Lie groupsG.

For the purposes of 2.11 we will make some assumptions onG. Wesuppose that there is a basis fora∗ such that relative to this basis therestricted roots ofG are of the form:±ti ± t j ,±ti and±2ti . We also assumethat the multiplicities of the restricted roots are as follows: fori < j , ti − t jhas multiplicitya and ti + t j has multiplicityb independent ofi < j ; therestricted rootsti have multiplicityc1, and the restricted roots of the form 2tihave multiplicityc2 independent ofi . We setc = c1 + c2. We assume thata = b. We will write G = G(a,b, c) to indicate that all these conditionshold. Here are the relevant examples. IfG = SU(ν + νo, ν), ν ≥ 2, andνo ≥ 1, thena = b = 2, c = 2νo + 1, andG satisfies the conditionc ≥ 2wheneverνo ≥ 1. If G = Sp(ν + νo, ν) thena = b = 4, c = 2νo + 3. IfG = SO∗(2n),n = 2ν + 1, thena = b = 4, c = 3. If G is of type EIIIthena = b = 6, c = 9. For the rank one cases, whenG = SU(1+ νo,1)a = b = 0 andc = 2νo + 1 ≥ 2 for νo ≥ 1; and whenG is of typeFII a = b = 0 andc = 15. Each of these groups can then be denoted byG = G(a,b, c) and the additional conditions ona,b, c are satisfied.

We may assume that there is a positive system ofh roots such that thecorresponding restricted roots agree with∆+(G(n)) except for multiplici-ties.

We suppose that any elementx in WP acts by sending anyti to ±tσ(i)whereσ is a permutation inSn. Therefore there is a group homomorphismF : WP → G(n). The second notion of “length ”of an elementx ∈ G(n)that was introduced above takes into account the multiplicitiesa = b, c.For any elementx = F(w) in the image of the homomorphismF, L2(x)


will therefore agree with the length ofw computed inWP. This is becausethe number of positive restricted roots ofG that become negative relativeto ∆+(G(n)) is also the number of positive roots ofh that become negativewith respect to a positive system ofh roots. The length of an elementin WP corresponding tox is then simply given byaR1 + bR2 + cR3 =a(R1 + R2)+ cR3.

(2.11) Proposition. Let G = G(a,b, c) be as above witha 6= 1 andc≥ 2.Then for anyw ∈ WP there is now′ ∈ WP such thatw→ w′.

Proof. If w → w′ in WP, then alsow → w′ in WPmin, and necessarilyw′ = sβw for some reflectionsβ associated to anh root β. This rootβmust be real but it is not necessarily simple. Now,sβ acting on theR vectorspace spanned by theti , i = 1, ...,n, has the formTβ as in A.5 and, usingA.5, F(sβ) = sα whereα ∈ ∆+(G(n)) represents the real rootβ. Thus it isenough to show that for any such reflectionsα in G(n) andx ∈ G(n) wehave thatL2(sαx)− L2(x) has absolute value at least 2.

Forx, y fixed let∆Ri be the changeRi (yx)−Ri (x) in the numberRi thatresults when applyingy on the left. We will consider onlyy = sα. We nowhave proposition A.4 which implies that∆R1+∆R2 and∆R3 cannot havedifferent signs. Therefore|L2(sαx)−L2(x)| = a (∆R1+∆R2)+c∆R3| ≥ 2as follows trivially whena = 0, and whena 6= 0 because this is at leastmin{a, c}. utsl(2) calculations.

We considerG = SL(2,R), K = SO(2), H the split Cartan{diag(r,r−1) : r ∈ R∗}. Let±α be the real roots relative toH, and letσ = σα be thecorresponding involution. There are two inequivalent principal series mod-ules forSL(2,R) (or Ad SL(2,R)) having trivial infinitesimal character andcontaining the trivial representation as subquotient. LetM denote the onethat contains the trivial representationC as submodule. IfD+, andD− de-note the two discrete series quotients ofM, thenσ interchangesD+, andD−.Let M be the principal series module containing the trivial representationas a quotient. The notation(. . .) when applied to Lie algebra modules shalldenote a notion of duality that exists in the categories of Harish–Chandramodules or categoryO modules with the same (generalized) infinitesimalcharacter as the trivial representation. In the concrete case ofsl(2) whichconcerns us in (2.12) and (2.13) this is obtained by taking the contragredientmodule and twisting the action with the involutionσ .

(2.12) Lemma. Up to isomorphism, there is a unique Harish–ChandramoduleZ for SL(2,R) with trivial infinitesimal character, such that

1) Z has the same composition factors (with multiplicity) asM2) Zσ ∼= Z3) φαψα(Z)→ Z is surjective ;

namely,Z ∼= M.


Proof.There are very few possible modulesZ with which to start. The list ofmodulesZ satisfying 1) in (2.12), and the corresponding indecomposablessummands ofZ is the following (exact sequences listed do not split)

a) Z = V+ ⊕ D−, with 0→ D+ → V+ → C→ 0b) Z = V+ ⊕ D− with V+ dual ofV+c) Z = V− ⊕ D+ ,with, 0→ D− → V− → C→ 0

d) Z = V− ⊕ D+, V− dual ofV−e) Z = Mf) Z = Mg) Z = F+ , 0→ V+ → F+ → D− → 0.h) Z = F− = F+ ,the dual ofF+.i) Z = C⊕ D+ ⊕ D−.

Considering all of all these, onlyM, M and C ⊕ D+ ⊕ D− are fixedby σ becauseσ interchangesD+ with D−. Moreover, onlyM satisfies thecondition 3). This last claim follows from (2.5), in the case ofSL(2,R),together with the fact thatψαC = 0, when applied to each of the differentlisted modules. For example the moduleC⊕D+⊕D− is discarded because3) fails forC. ut

LetG = SL(2,R)± the group of two by two real matrices of determinant±1 and K = O(2). Let M′ denote a principal series module with a onedimensional representationC as unique submodule. The dualM′ is thena principal series module withC as unique irreducible quotient. Note thatasG is disconnected it has two distinct one dimensional representations.

(2.13) Lemma. G = SL(2,R)± has a unique Harish–Chandra moduleZwith the same character asM′, and such thatφαψα(Z)→ Z is surjective;namelyZ ∼= M′.

Proof. We omit this proof because it is similar to and easier than the proofof Lemma 2.12. ut(2.14) Lemma. Let g be the Lie algebra ofSL(2,C). Then there is onlyone categoryO moduleZ such thatZ has the same character as a Vermamodule of highest weight zero,and such thatφαψα(Z)→ Z is surjective. Inthis caseZ is the dual of a Verma module.

Proof.This is immediate from Lemma 2.12. ut

3. Then-homology complex and asymptotically large representations

Recall that we are examining theG action on an analogue of the De Rhamcomplex. As is now classical, the Betti numbers ofG/P are related to the


trivial K module. Unfortunately there are not enoughK -invariant forms inthe De Rham complex to obtain the complete cohomology ofG/P. Howeverthe representation theory ofG provides a substitute for invariant forms.In the category of Harish–Chandra modules with a trivial infinitesimalcharacter (compatible(g, K )-modules of finite length and the action of thecenter of the enveloping algebra is by scalars as in the trivial representation),the functorX → (X/nX)H , the H-invariant part of the homology groupH0(n, X), is a covariant exact functor. This functor detects in a Harish–Chandra moduleX the trivial representation as a subquotient (see (3.1)).

We may generalize this and consider instead the vector space(X/nX)0.These are theh-invariant vectors inX/nX. SinceHo, the connected com-ponent of the identity ofH, acts trivially on this vector space, we obtain anH/Ho action. We call an irreducible Harish–Chandra moduleX with trivialinfinitesimal characterasymptotically largeor just large if (X/nX)0 6= 0.The asymptotically large representations are known to agree exactly withthe irreducible modules{π(δ)} of the Introduction whereδ, viewed asa Beilinson–Bernstein parameter (Langlands classification in [BB]), is at-tached to the openKKK orbit. These are Langlands submodules of principalseries modules{πδ}, induced from a minimal parabolic subgroup ofG.Thus every large representationπ(δ) gives rise to a characterζ(δ) of H/Ho.Decomposing intoH/Ho-isotypic components we have(X/nX)0,ζ(δ) theζ(δ) isotypic component of the 0-weight space inX/nX. We will applythese functors to the complexesI ∗(BGG) and J∗(BGG). In this way oneobtains a complex of finite dimensional vector spaces, with a basis whichis roughly given byWPPP, that also computesH∗(G/P;C). We begin witha well-known result which can be gleaned from [HS] for example, fromTheorem 3.6 there and most specifically (6.5).

(3.1) Lemma. For any irreducible Harish–Chandra moduleX, Hq(n;X)H = 0 for all q, unlessX = F, the trivial representation. IfX = F,then H0(n; F)H = C, and Hq(n; X)H = 0, if q 6= 0. For any irreducibleHarish–Chandra moduleX, and a parameterδ of a large representationπ(δ), Hq(n; X)0,ζ(δ) = 0 unlessX is isomorphic toπ(δ). If X is isomorphicto π(δ), thenH0(n; X)0,ζ(δ) = C, and Hq(n; X)0 = 0, if q 6= 0.

We remark that in any of the following results, alarge representa-tion π(δ) can replace the trivial representation, providedHq(n; . . . )0,ζ(δ)replacesHq(n; . . . )H . With these replacements the proofs can be modifiedmutatis mutandis except for (3.5) whereH0(n; X∗)H has to be replaced withH0(n; X∗)0.

(3.2) Lemma. Let X be a Harish–Chandra module such thatF, the trivialmodule, occurs exactly once as subquotient. ThenHq(n; X)H = 0 if q 6= 0,H0(n; X)H = C.


Proof.This follows from (3.1) and then induction on the number of compo-sition factors , using the long exact sequence forn-homology. We have threecases:A→ X → F, F → X → A, A→ X → B with A, B both 6= F.We prove the result for the first case. The other cases being done similarly.Suppose thatA→ X→ F is a short exact sequence. Then we have

· · · → Hq(n; A)→ Hq(n; X)→ Hq(n; F)→ Hq−1(n; A)→ · · ·

By induction on the length,Hq(n; A)H = 0 for all q. Thus if q 6= 0,Hq(n; X)H = 0. If q = 0, we obtain insteadH0(n; X)H = H0(n; F)H = C.

ut(3.3) Lemma. If X is a representation of the formI(w)or J(w) occurring inI ∗(BGG) or J∗(BGG), thenH0(n; X)H = 0 unless the trivial representa-tion occurs inX as subquotient. IfF is a subquotient ofX, thenH0(n; X)H

is one dimensional.

Proof.Any such moduleX containsF at most once. The result then followsfrom Lemma 3.2. ut(3.4) Lemma. Let X, Y be two Harish–Chandra modules of finite length.Assume that each of them containsF, the trivial module, exactly once asa subquotient. Then ag-module mapf : X→ Y has in its image the trivialrepresentationF as a subquotient, if and only ifH0(n; X)H → H0(n;Y)H

is an isomorphism.

Proof. We have an exact sequence 0→ Ker→ X → Y → Coker→ 0.Now on Harish–Chandra modulesX → X/(nX)H is exact. Hence we obtainan exact sequence 0→ H0(n;Ker)H → H0(n; X)H → H0(n;Y)H →H0(n;Coker)H → 0. If f does not containF in its image then by Lemma(3.2), H0(n;Ker)H = C and H0(n;Coker)H = C. Indeed, for Ker andCoker must containF as subquotient. Now returning to the exact sequence,one concludes thatH0(n; X)H → H0(n;Y)H is zero.

If on the other handf(X) containsF, we haveH0(n;Ker)H = 0 =H0(n;Coker)H , and again using the exact sequence (everything is at mostone-dimensional) we must have an isomorphism. ut

These features shared byI ∗(BGG) and J∗(BGG) suggest we considera chain complexX∗ of Harish–Chandra modules with trivial infinitesimalcharacter and finite dimensional cohomology

· · · →⊕

j

Xk, j →⊕

l

Xk+1,l → · · · .

for which the differential is obtained from a family of mapsf(k, j, j ′) :


Xk, j → Xk+1, j ′ (some of them possibly zero) and for eachk, a matrix(r j, j ′),such that the differential in the chain complex isr j, j ′ f(k, j, j ′). We assumealso thatF, the trivial module, occurs at most once as a subquotient of eachXi, j and letδk, j, j ′ be 1 or 0 according to whether or notf(k, j, j ′)(Xk, j )containsF as subquotient.

FromX∗ we define a second chain complex of finite dimensional vectorspaces by applyingH0(n; . . . )H to X∗.

(3.5) Proposition. The chain complexesX∗ and H0(n; X∗)H are quasi-isomorphic as complexes of vector spaces, via a chain map fromX∗ toH0(n; X∗)H .

Proof. Let X, Y, and Z be three consecutive terms in the complexX∗,eachX, Y, Z a sum of Harish–Chandra modules each having F at mostonce as subquotient. We have a short exact sequence defining one of thecohomology groups

0→ Im (X→ Y)→ Ker(Y→ Z)→ Ker/Im → 0 .

Apply the exact functorH0(n; . . . )H to obtain

0→ H0(n; Im (X→ Y))H → H0(n;Ker(Y→ Z))H

→ H0(n;Ker/Im )H → 0.

Recall that there are maps from the first exact sequence to the second sinceH0(n; . . . ) is a quotient(. . . )/n(. . . ).

For any irreducible moduleR, let [R : X] denote the multiplicity ofRin the semisimplification ofX. Then [R : Ker/Im ] = [R : Ker(Y→ Z)]−[R : Im (X → Y)]. From the conditions onX∗ we have that Ker/Im isfinite dimensional and that only the trivial representation, F, may contributeto cohomology, all other irreducible subquotients being infinite dimensional.Therefore,

dimKer/Im = [F : Ker(Y→ Z)] − [F : Im (X→ Y)].By Lemma (3.2),

dimH0(n;Ker/Im )H = dimH0(n;Ker(Y→ Z))H

− dimH0(n; Im (X→ Y))H

= [F : Ker(Y ∈ Z)] − [F : Im (X→ Y)].

Thus the complexH0(n; . . . )H has the same cohomology as the com-plex X∗. ut

As the complexesI ∗(BGG) ∼= J∗(BGG)[−n] satisfy the conditions ofProposition 3.5, we obtain


(3.6) Proposition. The complexJ∗(BGG)[−n] is quasi-isomorphic toa complex of finite dimensional vector spaces of the form

· · · −→⊕σ∈WPPP

l(σ)=k

H0(n; J(χ(ξ(σ)))H −→⊕τ∈WPPP

l(τ)=k+1

H0(n; J(χ(ξ(τ)))H −→ · · ·

whose cohomology computesH∗(G/P;C).Using large representations instead of the trivial representation the fol-

lowing statement can also be proved. The proof is almost identical to theproof of (3.6) and so is omitted.

(3.7) Proposition. The complexJ∗(BGG)[−n] is quasi-isomorphic toa complex of finite dimensional vector spaces of the form

· · · −→⊕σ∈WPPP

l(σ)=k

H0(n; J(χ(ξ(σ))))0 −→⊕τ∈WPPP

l(τ)=k+1

H0(n; J(χ(ξ(τ))))0 −→ · · ·

whose cohomology computesH∗(G/P;C).Remark.There is a critical distinction between the chain complexes in (3.6)and (3.7) even though both compute the cohomology ofG/P. In example(8.1) and the one in the Introduction, the terms arising from (3.6) are labeledin a box. However to compute torsion in the cohomology one must use termsarising from (3.7) that are not boxed.

Remark.If G andP are complex groups, Kostant’s well-known result statesthat the complex in (3.6) computesH∗(G/P;C) with differentials the zeromap. In this case of complex groups there is no torsion.

4. Cellular resolutions

Cellular chain complexes and resolutions.

There is a natural relationship between the previous complexes obtainedfrom representation theory and standard complexes available from topology.To see this, in this section we describe two well-known methods to computethe cohomology of a special type of stratified topological space. We takecoefficients in a principal ideal domain,R.

Let Z be a Hausdorff topological space, and assume thatZ has a filtration

∅ ≡ Z−1 ( Z0 ( Z1 ( · · · ( Zm = Z

where eachZi is closed in Z and Z\Zm−1 is open and dense. We donot assume that the stratumSk =Zk\Zk−1 is a union of cells. Under the


conditionHk(Zp, Zp−1; R) = 0 if k 6= p, there is a chain complex denotedCell({Zk}; R):

· · · → H p(Zp, Zp−1; R)→ H p+1(Zp+1, Zp; R)→ · · ·(4.1)

that computesH∗(Z; R) (e.g. [Mu] Theorem 39.4).Under the weaker conditionHk(Zp, Zp−1; R) free = 0 if k 6= p, there is

also a useful chain complex denoted Cell({Zk}; R) free:

· · · → H p(Zp, Zp−1; R) free→ H p+1(Zp+1, Zp; R) free→ · · ·(4.1′)

The differentials are obtained from the long exact sequences of the pairs(Zp+1, Zp) and(Zp, Zp−1). Indeed these sequences give maps

dp : H p(Zp, Zp−1; R)→ H p(Zp; R)dp+1 : H p(Zp; R)→ H p+1(Zp+1, Zp; R).

The differential in the complex Cell({Zk}; R), d : H p(Zp, Zp−1; R) →H p+1(Zp+1, Zp; R), is just the compositiond = dp+1 ◦ dp.

This chain complex has an alternate formulation in terms of sheaf co-homology which has the added advantage of not needing the vanishingresult but at the cost of using a double chain complex. We shall use it in thisform as well as in the simplified form that results when the vanishing resultis available.

Recall thatSp is the stratumZp\Zp−1, and thatSp is open inZp. Let

j p : Sp→ Zp

i p−1 : Zp−1→ Zp

be the inclusions. We shall use repeatedly the fundamental exact sequenceof sheaves ([Iv] II.6.11)

0→ j p! j ∗pA→ A→ {i p−1}∗i ∗p−1A→ 0(4.2)

whereA is any sheaf onZp, to define inductively sheaves onZ that weshall denoteRSp.

To begin, recall that the constant sheafR on Zm is p∗mR where pm :Zm→ {pt}. Sincepm−1 = pm ◦ jm mapsSm to {pt}, the constant sheaf onSm is p∗m−1R= j ∗m ◦ p∗mR. Then, in context,R unambiguously will denotea constant sheaf. We define a sheaf onZ = Zm by

RSm = jm!R = jm! j ∗mR.

From ([Iv] IV.8.1) we see that

H∗(Zm, Zm−1; R) ∼= H∗(Zm;RSm).

To continue, use (4.2) withi ∗m−1R the sheaf onZm−1, obtaining

0→ { jm−1}! j ∗m−1i∗m−1R→ i ∗m−1R→ {im−2}∗i ∗m−2i

∗m−1R→ 0.


SinceZm−1 is closed,{im−1}∗ = {im−1}! and the functor{im−1}∗ is exact;applying it to this sequence we obtain another short exact sequence, say(4.2′). Let jm−1 = im−1 ◦ jm−1 be the inclusion ofSm−1 into Z, and definea sheaf onZ by

RSm−1 = { jm−1}!R = { jm−1}!{ jm−1}∗R.Again we have that

H∗(Zm−1, Zm−2; R) ∼= H∗(Zm−1; { jm−1}!RSm),

while ([Iv] II.5.4) gives

H∗(Zm−1; { jm−1}!RSm)∼= H∗(Zm; {im−1}∗{ jm−1}!R)= H∗(Zm; { jm−1}!R)= H∗(Zm;RSm−1) .

Iterating this procedure we obtain the sheavesRSp on Z whose hyper-cohomology computes the relative cohomology groups.

In order to obtain the differential, in the usual way we shall iden-tify a sheaf with a graded complex of sheaves. There is then a trans-lation functor, so that we have for exampleRSm−1[−1]. Composition ofthe mapRSm−1[−1] → {im−1}∗i ∗m−1R[−1] with the connecting morphismim−1∗i ∗m−1R[−1] → RSm gives a map

RSm−1[−1] → RSm.

Since we will be interested in complexes up to quasi-isomorphism, it willbe more convenient to operate in the derived category. The procedure to gofrom short exact sequences of sheaves (or graded complexes of sheaves) todistinguished triangles in the derived category of complexes of sheaves ofR-modules is described in [Bo1] § 5. Each of the exact sequences (4.2) thuscorresponds to a distinguished triangle and from the collection of these wederive the sequence

· · · → RSp[−1] f p−→ RZp[−1] δp+1−→ RSp+1

f p+1−→ RZp+1

δp+2−→ RSp+2[1] → · · ·We obtain a mapd : RSp[−1] −→ RSp+1 by settingd = δp+1 ◦ f p. Sincef p+1 andδp+1 are from the same distinguished triangle,f p+1 ◦ δp+1 = 0.Thusd ◦ d = δp+2 ◦ f p+1 ◦ δp+1 ◦ f p = 0. Hence in the case at hand, weobtain in this way a diagram of objects and maps in the derived category

0→ · · · → RSp[p−m] → RSp+1[p−m+ 1] →RSp+2[p−m+ 2]→ · · · → RSm → R .


In particular each mapRSp[−1] → RSp+1 which results from the compo-sition

RSp[−1] → RZp[−1] → RSp+1,

corresponds to a map in the composition

H•(Zp, Zp−1; R)→ H•(Zp; R)→ H•(Zp+1, Zp; R)[1].In fact, the above gives a differential,d, in a double chain complex

H•(Zp, Zp−1; R)→ H•(Zp+1, Zp; R)[1].(4.3)

In this case one has a double chain complex whose hypercohomology, thecohomology associated to the associated single chain complex, computesH∗(Z; R). This will be used in Lemma 4.13 III). In the presence of thevanishing conditionH j (Zp, Zp−1; R) = 0 wheneverj 6= p, this doublechain complex simplifies and we indeed obtain then the differentiald ofCell({Zk}; R).(4.4) Definition. A {Zp} cellular resolution of the constant sheaf is a dia-gram of objects and maps in the derived category,

· · · −→ RSp[p−m] d−→ RSp+1[p−m+1] d−→ RSp+2[p−m+2] −→ · · ·parametrized by the strata as described above, and satisfyingd ◦ d = 0.

Remarks4.5. We have refrained from calling the diagram a chain complexbecause the derived category is not an abelian category. Also, the term“cellular resolution” is always with respect to a filtration, and it is definedindependently of the vanishing conditionHk(Zp, Zp−1; R) = 0 if k 6= p.

act∗-acyclic Cellular Resolutions.

Recall thatGGG is a complex group, a complexification ofG. Similarlywe let KKK , PPP, LLL, etc. denote subgroups ofGGG that are complexifications oftheir corresponding real groups.We may assume that the involutionθ hasbeen extended toGGG so thatKKK is the fixed point set of the involution. ThegroupPPP has a Levi-decompositionPPP = LLLUUU, andθ(LLL) = LLL, θ(UUU) = UUU∗.Let BBB denote the Borel subgroup with Lie algebrab, and denote the usualflag manifoldGGG/BBB by XXX. One may identifyXXX with the variety of Borelsubgroups inGGG.

Recall that the generalized flag manifoldX = G/P ' K/K ∩ P. Weshall useXC for the quotient spaceKKK/KKK ∩ PPP= KKK/KKK ∩ LLL, and we shall referto XC as a complexification ofX or as the openKKK -orbit in GGG/PPP.

(4.6) Proposition. The natural inclusion ofX into XC is a homotopyequivalence. HenceHq(X; R) ' Hq(XC ; R).


Proof. The inclusion map of the groupK into KKK (resp.K ∩ L into KKK ∩ LLL)is a homotopy equivalence sinceK (resp.K ∩ L) is a maximal compactsubgroup in its complexificationKKK (resp.KKK ∩ LLL). The fibrations

1−→ K ∩ L −→ K −→ K/K ∩ L −→ 1

1−→ KKK ∩ LLL −→ KKK −→ KKK/KKK ∩ LLL −→ 1

give rise to long exact sequences of homotopy groups. AsK andKKK (resp.K∩L andKKK∩LLL) have isomorphic homotopy groups, one concludes from thefive lemma that the inclusion ofX into XC is a weak homotopy equivalence.Since the groups are Lie groups, these spaces are CW complexes; hence,by Whitehead’s theorem, the weak homotopy equivalence is a homotopyequivalence. utRemark.The preceding argument easily extends to any finite index subgroupof the set of real points ofGGG. One just observes that each connected compon-ent of KKK/KKK ∩ LLL is homotopy equivalent to the corresponding componentof K/K ∩ L.

The complexificationXC sits as the typical fiber in a natural fibrationover XXX. Indeed, letKKK ×

KKK∩PPPXXX be the quotient ofKKK × XXX by the two-sided

action of KKK ∩ PPP of the form l · (k, x) = (kl−1, lx). Then KKK ×KKK∩PPP

XXX is

a complex manifold with dimC KKK ×KKK∩PPP

XXX say d. There is a natural map

act: KKK ×KKK∩PPP

XXX → XXX given by act([k, x]) = kx, with fibers diffeomorphic

to XC :

XC −→ KKK ×KKK∩PPP

XXXyact

XXX

(4.7)

(4.8) Lemma. For anyx in XXX, (Rqact!(R))x ∼= Hqc (XC ;R).

Proof.The stalks(Rqact!(R))x are isomorphic toHqc (act−1(x);R)([Iv] VII

1.4). Since the fibers of this map are of the formKKK/KKK ∩ PPP, the resultfollows. ut(4.9) Lemma. For anyx in XXX, (Rqact∗(R))x ∼= Hq(XC ;R).Proof. Since the typical fiberXC is homotopy equivalent toX, a compactspace, andR is a commutative ring, the result follows from [Iv] IV 1.6.ut(4.10) Remark.As in ([Bo1] VI 2.8), one calls the sheafR act!-acyclic ifRqact!(R) = 0, for q ≥ 1. We shall define a variation of this for act∗ of


a cellular resolution ofKKK ×KKK∩PPP

XXX adapted to a particular filtration. But first

we need to describe the filtration.

The groupPPP acts onXXX and has orbits parametrized byWPPP. This givesa filtration {XXXk} of XXX consisting of unions ofPPP orbits; namely,XXX0 is thesmallest dimensional orbit,PPP/BBB, of dimension saym, andXXXk is the unionof all PPP orbits of dimension less than or equal tom+ k. Since the closure ofa PPP-orbit consists of lower dimensionalPPP-orbits, XXXk is closed. The stratathen consist of the union ofPPP-orbits of dimension exactlym+ k. By meansof the fibration (4.7), one can define compatible filtrations ofKKK ×

KKK∩PPPXXX

andXC .For KKK ×

KKK∩PPPXXX, the pull-back, via act∗, of {XXXk} gives a filtration denoted

by {ZZZk}, whereZZZk consists of sets of the formKKK ×KKK∩PPP

SSS, SSS a PPP-orbit of

dimension at mostm+ k. The strata consist of sets of the formKKK ×KKK∩PPP

SSS,

SSS a PPP-orbit of dimension exactlym + k. Cellular resolutions of a sheafon KKK ×

KKK∩PPPXXX shall always be with respect to this filtration without further

comment.For XC , let x be in the openKKK -orbit in XXX and assume thatx corresponds

to a Borel subgroupBBB contained inPPP. Let f : XC −→ KKK ×KKK∩PPP

XXX be the

inclusion into the fiber act−1(x). Then the pull-back, viaf ∗, of {ZZZk} givesa filtration of XC denoted{Yp}, i.e. {Yp = f −1{ZZZp ∩ act−1(x)}}. The nextresult follows readily from the definitions.

(4.11) Lemma. If A• is a {ZZZp}-cellular resolution of the constant sheafRon KKK ×

KKK∩PPPXXX, then f ∗A• is a {Yp}-cellular resolution of the constant sheaf

R on XC .

(4.12) Definition.A {ZZZp}-cellular resolutionA• = {Ap} of R on KKK ×KKK∩PPP

XXX

is said to be anact∗-acyclic cellular resolution if for eachj and for xin the openKKK-orbit in XXX, among the stalks of the derived sheaves{Hq(act∗(A j )[n])x}, the only non-zero cohomology group isH0(act∗(A j )[n])x which is assumed to be torsion free. Recall thatn = dimC XC = dimRX.

Fix a prime p. A {ZZZp}-cellular resolutionA• = {Ap} of Z on KKK ×KKK∩PPP

XXX

is said to be ap − act∗-acyclic cellular resolution if for eachj andfor x in the openKKK -orbit in XXX, among the stalks of the derived sheaves{Hq(act∗(A j )[n])x}, we haveHq(act∗(A j )[n])x, free = 0 if q > 0. Inaddition we assume that there is a chain map from the chain complex· · · → H0(act∗(A j )[n])x, free→ H0(act∗(A j+1)[n])x, free→ · · · to a sec-ond complex· · · → Bj → Bj+1→ · · · where

a) Bj is a freeZ-module of the same rank asH0(act∗(A j )[n])x, free


b) for each j there aredi 6= 0modpso that each mapH0(act∗(A j )[n])x, free

→ Bj is of the form(n1,n2, ...,nr )→ (d1n1,d2n2, ...,dr nr )

c) the compositionH•(act∗(A•)[n])x → H•(act∗(A•)[n])x, free→ B• ofchain maps induces an isomorphism in cohomology.

(4.13) Lemma. I) If A• is anact∗-acyclic cellular resolution of the constantsheafR, then

a) the chain complex ofR-modules

· · · → H0(act∗(A j )[n])x→ H0(act∗(A j+1)[n])x→ · · ·computes the cohomologyH∗(XC ; R). Moreover, this chain complexagrees with the cellular chain complexCell({Yp}; R) of XC associatedto the filtration{Yp};

b) in particular, for coefficientsZ, the chain complexH∗(XC ; f ∗A•) isa cellular chain complex having cohomologyH∗(XC ;Z).II) If A• is a p− act∗-acyclic cellular resolution of the constant sheaf

Z, then for eachj there is a surjective map between the j-th cohomology of

· · · → H0(act∗(A j )[n])x, free→ H0(act∗(A j+1)[n])x, free→ · · ·and H j (XC ;Z) whose kernel consists of torsion classes annihilated byintegers not divisible byp.

III) If A• is a cellular resolution of the constant sheafZ, then thechain complexH•(XC ; f ∗A•) has hypercohomologyH∗(XC ;Z). There isa double chain complex of the form:R = · · · → Rj → Rj+1→ · · · whereeachRj is itself a chain complex quasi-isomorphic toH•(XC ; f ∗A j ). Thecohomology of the associated single complexR is H∗(XC ;Z).

IV) H j (Yp,Yp−1;Z) free= 0 if j 6= p.

Proof. The first part of our argument applies equally to all the statementsI), II), III) and IV), and assumes only thatA• is a {ZZZp}-cellular resolutionas onKKK ×

KKK∩PPPXXX as in 4.4. Hence we will first prove III). Then this with

vanishing conditions will yield I) and II). The proof of statement IV usesthe first part of the argument but is best understood after the material from§ 5 is introduced. So the proof will be deferred until (5.14).

Fix x in the openKKK -orbit (diffeomorphic toKKK/KKK ∩ BBB) in XXX, and letW containingx be an open contractible set in this openKKK -orbit. There isthe principal fibrationr : KKK → KKK/KKK ∩ BBB. SinceW is contractible we cantake a cross-section tor over it,r # : W→ KKK , with r #(x) = e. Also we mayassume that the fibrationr trivializes overW. Consider the spaceW× XCwith filtration {W×Yp}, and letπ : W×XC → XC be the projection. LetBbe any sheaf onXC . SinceW is contractible, the Vietoris–Begle Theorem([Bo1] V, 3.13) says thatH∗(XC ;B) ∼= H∗+n(W×XC ;π∗B), in particularH∗(XC ; R) ∼= H∗+n(W× XC ; R).


We shall show thatW×XC is diffeomorphic to act−1(W) as spaces withfiltrations.This will be used to prove the statement in a) by reducing thecomputation of the cohomology ofXC to the computation of the cohomo-logy over act−1(W). For the rest of the proof one need not require thatWbe contractible.

Fix SSSa PPP-orbit in XXX and letZZZS= KKK ×KKK∩PPP

SSSbe the associated stratum in

KKK ×KKK∩PPP

XXX. Clearly act−1(x) = {[k, k−1x]modKKK ∩ LLL : k ∈ KKK} and act−1(x)∩ZZZS = {[k, k−1x]modKKK ∩ LLL : k ∈ KKK and k−1x ∈ SSS}. Similar statementshold for act−1(W) and act−1(W) ∩ ZZZS. We claim that act−1(W) ∩ ZZZS ≈W× (ZZZS∩ act−1(x)). For this, consider the map

F : W× (ZZZS∩ act−1(x)) −→ KKK ×KKK∩PPP

XXX

given by(w, [k, k−1x]modKKK ∩ LLL) → ([r #(w)k, k−1x]modKKK ∩ LLL). Giventhe choice ofr # and thatr trivializes overW, we can identify the image ofF with the set{[k, k−1w]modKKK ∩ LLL : k ∈ KKK , w ∈ W and k−1w ∈ SSS

}.

However this is just act−1(W) ∩ ZZZS. Similarly, act−1(W) ≈ W× act−1(x).As A• is a ZZZp-cellular resolution ofR, we examine its restriction to

act−1(W)with theZZZS∩act−1(x) filtration. Letg: ZZZS∩act−1(x)→ act−1(x)be the inclusion of a typical strata. Since act−1(W) ≈ W×act−1(x) andA•is cellular, eachA j restricted to act−1(W) is quasi-isomorphic to a complexconsisting in each degree of a direct sum of sheaves of the formRW⊗g!RS′,SSS′ = ZZZS∩act−1(x) (with a shift in dimension that we have omitted). AsA•is {ZZZp}-cellular resolution of the constant sheafR on KKK ×

KKK∩PPPXXX, andW is

contained in the openKKK -orbit, then

H∗(act−1(W);RW ⊗ g!RS′

) = H∗(W× (ZZZS∩ act−1(x)

) ;RW ⊗ g!RS′)

= H∗((KKK ×

KKK∩PPPSSS∩ act−1(x)); g!RS′

).

This argument common to I), II), III), IV) is all one needs to obtain III). Thedouble chain complex obtained is of the same form as the one describedin (4.3).

If we assume the conditions in I) then as already observed, by computingcohomology on act−1(W) we have obtained the cohomology that is neededfor a). This time we obtain a chain complex of the form (4.1) on act−1(x),that computes the cohomologyH∗(XC ; R).

To prove II, we use from the proof of I that the double chain com-plexH•(act∗(A•)[n])x has hypercohomologyH•(XC ;Z). From the defin-ition of p − act∗-acyclic cellular resolution, there are two chain com-plexes of freeZ-modules of finite rank –H•(act∗(A•)[n])x, free and B• –and the first injects into the second, term by term, via a chain mapT.


That there is a surjective map from the cohomology ofH•(act∗(A•)[n])xto H∗(XC ;Z) follows easily from III and the assumption that the com-position H•(act∗(A•)[n])x → H•(act∗(A•)[n])x, free → B• induces anisomorphism in cohomology. We use ambiguouslyT for the chain mapand the induced map on cohomology. We examine the kernel of thismap in cohomology. So letc be a cycle inH0(act∗(A j )[n])x, free andsuppose thatT([c]) = 0. Then there is anx such thatdx = T(c), al-thoughx need not be in the image ofT. However, asT can be written asT(n1, ...nr ) = (d1n1, ..,dr nr ), there is an integern, a product of all thedi ,such thatnx = T(y). Now dT(y) = dnx = nT(c) and we obtain thatn[T(c)] = 0. The integern is not divisible byp becausep does not divideany of the numbersdi that arise. ut

Cellular resolution ofXC .

The next results are used to show that we may computeH∗(G/P;Z) inthis way and to relate it to the usual Bruhat cellular decompositions. Recallfrom (4.6) that in order to compute the cohomology ofX it is enough towork with XC .

(4.14) Proposition. Fix any x in the openKKK -orbit in XXX. For w ∈ WPPP letSSS= PPPwBBB be aPPP-orbit in XXX. Via the identification ofact−1(x) with XC , theopenKKK orbit in GGG/PPP, act−1(x) ∩ KKK ×

LLL∩KKKSSS is identified withXC ∩ NNNwPPP.

Proof.Recall that

act: KKK ×LLL∩KKK

XXX→ XXX.

We need to determine the intersection of the fibers act−1(x) with KKK ×LLL∩KKK

SSS,

whereSSS is aPPP-orbit. For this, letp: KKK × XXX→ KKK ×LLL∩KKK

XXX be the projection,

and notice that all fibers are bi-holomorphic toLLL∩KKK . So forA andB subsetsof KKK ×

LLL∩KKKXXX, in order to determineA ∩ B it is enough then to determine

p−1(A∩ B), which equalsp−1(A) ∩ p−1(B).Rather than working withGGG/PPP, it will be more convenient to work with

PPP\GGG as will become clear below.Now the element(k, k−1x) is in p−1(act−1(x)∩KKK ×

LLL∩KKKSSS) precisely when

k is in KKK andk−1 ∈ pwBBB for somep in PPP andw a representative of a Weylgroup element. Hencek−1 ∈ PPPwBBB ∩ KKK . As (k, k−1x) is in the fiber ofp,we may modifyk by kl, l ∈ LLL ∩ KKK . Thenk−1 is similarly adjusted andsinceLLL ∩ KKK is contained inPPP and inKKK , it follows that the element mapsto PPPwBBB ∩ (LLL ∩ KKK\KKK), i.e. the intersection of theBBB-orbit with the openKKK -orbit in PPP\GGG. But theBBB-orbits in PPP\GGG are simplyNNN-orbits – hence theclaim. ut


(4.15) Proposition. Fix any x in the openKKK -orbit in XXX. For w ∈ WP letSSS= PPPwBBB be aPPP-orbit in XXX. Via the identification ofact−1(x) with XC , theopenKKK -orbit in GGG/PPP, act−1(x)∩KKK ×

LLL∩KKKSSSintersected withG/P is identified

with an N+ orbit in X = G/P.

Proof.From (4.14), act−1(x) ∩ KKK ×LLL∩KKK

SSScan be identified with the intersec-

tion of anNNN-orbit with the openKKK -orbit in PPP\GGG. In fact, as in the proof of(4.14), it suffices to compute the image inPPP\GGG of KKK ∩ PPPwBBB = KKK ∩ PPPwNNN,for somew ∈ WPPP.

Multiplying by PPP on the left will not change anything, so we mayconsider instead the setPPP[KKK ∩ PPPwNNN]. Now we must intersect this last setwith {PPPk : k ∈ KKK}. This is the inverse image of the openKKK -orbit in PPP\GGG.Each parabolic subgroupkPPPk−1 intersectsG in kPk−1. This intersectionwith G produces a bijection between cosets:{PPPk : k ∈ K} → { Pk : k ∈ K}and ∪

k∈KPk= G. It suffices to intersectPPP[KKK ∩ PPPwNNN] with G. Assume that

w has a representative inK , that is,w is a representative of an elementin WP. In this casePPPwNNN containsPwN+. We first verify that it cannotcontain PwN+ and Pw′N+ if w 6= w′ as elements inWP ⊂ WPPP. If itdoes, thenPw′N+ ⊂ PPPw′NNN. ThereforePPPwNNN ∩ PPPw′NNN 6= ∅. Notice thatWPPP classifiesPPP-orbits inGGG/BBB or BBB-orbits (= NNN-orbits) inPPP\GGG. ThereforePPPwNNN∩ PPPw′NNN 6= ∅ implies thatPPPwNNN = PPPw′NNN andw = w′ as elements inWP ⊂ WPPP. Therefore since the union of all the double cosetsPw′′N+ overWP is G, the desired intersection consists of exactly oneN+-orbit. ut

We conclude this section with the result that states that the filtration{Yp}of XC gives rise to a chain complex Cell({Yp};Z), hence we can computethe cohomology ofX with the sheaf-theoretic methods described earlier inthis section.

We introduce some notation for the proof of 4.16. The inclusion ofXpinto Yp gives rise to maps in cohomology:

h jp : H j (Yp,Yp−1;Z)→ H j (Xp, Xp−1;Z)

The Bruhat decomposition gives a cellular decomposition ofX, i.e.Hq(Xp, Xp−1;Z) = 0 for q 6= p, hence the mapsh j

p = 0 if j 6= p.Similarly, H p(Xp, Xp−1,Z) isZ- free, sohp

p = hp will factor as

H p(Yp,Yp−1;Z)→ H p(Yp,Yp−1;Z) free→ H p(Xp, Xp−1;Z).Denote byTp the mapH p(Yp,Yp−1;Z) free → H p(Xp, Xp−1;Z). Thesegive rise to a chain mapT between two chain complexes as in 4.1 and 4.1′.Using (4.13) IV, if we denote byS = Cell({Yk};Z) free there a chain com-plex determined as in 4.1’ byH p(Yp,Yp−1;Z) free and the correspondingdifferentials.

Since, by 4.13 III)H∗(XC ;Z) is quasi-isomorphic to

· · · → H∗(Yp,Yp−1;Z)→ H∗(Yp+1,Yp;Z)→ · · · ,


we have a factorization of the quasi-isomorphismh in terms ofT

H∗(XC ;Z) h−−−−→H∗(G/P;Z)↘↗ TS

Changing coefficients toR= Z/2Z there is also a factorization:

H p(Yp,Yp−1; R)→ H p(Yp,Yp−1; R) free⊗Z

R → H p(Xp, Xp−1; R).

ForZ/2Z coefficients instead of the integers, we have the following resultof Takeuchi [Ta].

Theorem. The real Schubert cells form aZ/2Z perfect cell decomposition,that is,

Hk(X;Z/2Z) =⊕w∈WP

l(w)=k

Z/2Z.

From this follows

(4.16) Proposition. If A• is any {ZZZp}-cellular resolution of the constantsheafZ on KKK ×

KKK∩PPPXXX, thenA• is 2− act∗-acyclic. In particular, there is

a surjective mapHr (S)→ Hr (G/P;Z). The kernel of the map consists oftorsion classes[c] such that for some odd integern, n[c] = 0.

Proof. Recall that sinceA• is any{ZZZp}-cellular resolution of the constantsheafZ then, using (4.11), we obtain a{Yp}-cellular resolution by appli-cation of f ∗. This in turn gives rise to a chain complex of the form (4.3)which by (4.13)III) computes the cohomology ofXC . This chain complexhas already been related to act∗(R) in (4.9). So it is enough to focus ourattention on a complex of the form (4.3) that also computes act∗(R)x wherex is in the openKKK orbit in XXX.

Recall that each cohomology groupH p(Xp, Xp−1;Z) is of the form⊕

l(ω)=pZ . Over R this then looks like ⊕

l(ω)=pR. The mapT is a direct sum of

expressions of the form⊕l(ω)=p

dp,ω for eachp, wheredp,ω is an integer. Thus

we have from (4.13) IV)

H p(Xp, Xp−1;Z)x ⊕l(ω)=p

dp,ω

H p(Yp,Yp−1;Z) freexH•(Yp,Yp−1;Z)


The composition of the two arrows induces the isomorphism in cohomologybetweenH∗(Y;Z) andH∗(G/P;Z).

We show that the integersdp.ω must be odd. To see this we repeat theconstructions above overR = Z/2Z. By Takeuchi’s theorem, each copy ofRthat occurs inH p(Xp, Xp−1;Z) ⊗Z R= ⊕

l(ω)=pR gives rise to a co-cyclec

which in turn gives rise to a non-zero cohomology class[c] of H∗(G/P;Z).Thus if an integerdp,ω is even, then the corresponding mapT (over R) willfail to be surjective in cohomology. We conclude that eachdp.ω must beodd. This finishes the verification thatA• is 2− act∗-acyclic. Here the roleof B• is played by the chain complex with the termsH p(Xp, Xp−1;Z). ut

Recall the chain complexS = Cell({Yk};Z) free determined byH p(Yp,Yp−1;Z) free and the corresponding differentials.

(4.17) Corollary. If A• is a{ZZZp}-cellular resolution of the constant sheafZ,then Hr (S) and Hr (G/P;Z) have the same Betti numbers and the samenumber of summands of the formZ/2sZ.

Proof.As in the proof of (4.16) it suffices to consider the chain complexS.The kernel of the mapHr (S)→ Hr (G/P;Z) in (4.16) consists of torsionclassesc such that for some odd integern, nc = 0. ConsequentlyHr (S)and Hr (G/P;Z) have the same Betti numbers and the same number ofsummands of the formZ/2sZ. utRemark.We shall see in (9.9) that if Cell∗({Xp};Z) has a summand of theformZ/2sZ then necessarilyp= 2.

(4.18) Corollary. If A• is a{ZZZp}-cellular resolution of the constant sheafZ,then the differentialsdp of the chain complex:

Cell∗({Xp};Z) =· · ·→Hp(Xp, Xp−1;Z)→Hp−1(Xp−1, Xp−2;Z)→· · ·and the differentialsdA

p of A• can both be represented by matrices withentriesdp(x, y) , dA

p (x, y) x and y in WP. We then have thatdp(x, y) 6= 0if and only ifdA

p (x, y) 6= 0.

Proof. As in (4.17) this follows from the fact that the mapT in (4.16) isa chain map and therefore:dpTp+1 = TpdA

p . Also T is given by a diagonalmatrix of odd integers. ut

5. BGG resolution – sheaf version

In this section we present a sheaf theoretic formulation of the BGG complexdescribed in § 1. We show that it gives an act∗-acyclic cellular resolutionof the constant sheaf. Crucial to this reformulation is the correspondenceintroduced by Beilinson–Bernstein [BB] and Brylinski–Kashiwara [BK]


betweeng-modules and perverse sheaves. So we begin this section witha review of the relevant terminology.

Let V be a quasi-projective algebraic variety overC. Following [Bo1]§ 5 we let K(V, R) be the category with objects complexes of sheavesof R-modules whose cohomology sheaves are non-zero only in finitelymany degrees, and with morphisms homotopy classes of maps. We willconsiderD(V, R) the derived category with the same objects asK(V, R)but morphisms not necessarily chain maps but rather certain equivalenceclasses of maps fromK(V, R).

Usually we will fix a filtration ofV, {Vi }and then denote byD(V, {Vi }, R)the derived category of complexes of sheavesF • on V which are con-structible relative to the filtration{Vi}. This means that the stalks of all thecohomology sheavesH∗F • are finitely generatedR-modules and that theH∗F • restricted to eachVi are cohomologically locally constant.

One defines the category of perverse sheaves Per(V, {Vi},C), as anabelian subcategory inD(V, {Vi},C) satisfying the following conditions:

a) HqF • restricted toVi is non-zero only if−dimV ≤ q ≤ dimVi .

b) The Verdier dualDDDF • is a complex of sheaves also satisfying a).

There are perverse cohomology groupsp Hq(F •) (see [BBD]) associatedto any complexF • in Per(V, {Vi},C), and there is also an equivariantversion of these categories. Suppose thatSSS is an algebraic group with anaction onV preserving eachVi . We denote by PervSSS(V, {Vi},C) the categoryof perverse sheaves on(V, {Vi}) with trivial SSS-action.

Under the Riemann–Hilbert correspondence of Kashiwara and Mebkhout(see [Bo2]),F • corresponds to a complex of holonomicD-modules andconversely such aD-module corresponds to a perverse sheaf.

The functorP .

A Harish–Chandra (or(g, K )) module is a finite lengthg-module witha compatible action ofK (hence ofKKK ). We denote byHHHCCCρ the categorywith objects Harish–Chandra modules with a trivial infinitesimal characterand morphisms(g, K )-maps. There are several classifications of the simplemodules inHHHCCCρ (see [VoIII] Prop. 2.7). For our purposes the relevant oneis due to Beilinson–Bernstein. LetD be the set of pairs(Oi ,Li ), whereOiis aKKK -orbit in the flag manifoldXXX = GGG/BBB, andLi is aKKK -equivariant localsystem onOi (i.e. aKKK -equivariant line bundle with a flat connection).

Let OXXX denote the sheaf of germs of holomorphic functions on the flagmanifold XXX. The action ofGGG on XXX by holomorphic maps provides aU(g)action onOXXX. For anyg-moduleM, HomU(g)(M,OXXX) is a sheaf onXXX ofvector spaces with an action byKKK . If one takes a projective resolution ofM,one obtains an objectRHomU(g)(M,OXXX) in a derived category. ForM anyg-module letIρ, Iρ ⊆ Z(g), be the kernel of the infinitesimal character. Set


Aρ = U(g)/U(g)Iρ, and define

P (M) = RHomAρ (M,OXXX).

P (M) = DDD(P (M)[dimXXX])One shows that ifM is in HHHCCCρ, thenP (M)[dimXXX] is in PervKKK (XXX, {Oi },C),and this is the data attached toM by Beilinson–Bernstein.

A similar statement applies to modules in categoryO (the categoryof U(g)-modules with trivial infinitesimal character introduced by BGG).For w ∈ WPPP denote bySSS(w) the PPP-orbit PPPw−1BBB/BBB of GGG/BBB. This orbithas dimensionl(w) + m. For the category, denotedOPPP

ρ , of compatible(g,KKK ∩ PPP)-modules of finite length, locally finite aspmodules with trivialinfinitesimal character,P (M) is in PervKKK∩PPP(XXX, {SSS(w)},C).(5.1) Theorem. The functorM → P (M) establishes an equivalence ofcategories between

a) OPPPρ andPervKKK∩PPP(XXX, {SSS(w)},C);

b) HHHCCCρ andPervKKK (XXX, {Oi },C).BGG resolution (sheaf version).

The projective varietyXXX has the filtration{XXXk} where{XXXk} consists ofPPP-orbits of dimension≤ m + k and thePPP-orbits are parametrized byWPPP.The stratumXXXk\XXXk−1 is a disjoint union ofPPP-orbits. This more detailedstructure of the stratum permits a more refined analysis than arises typicallyfrom (4.2). So letSSS(w) be aPPP-orbit in the stratumXXXk\XXXk−1 with dw ≡dimC SSS(w) = m+ k, and let j(w) : SSS(w) → XXXk and j (w) : SSS(w) → XXXbe the inclusions described in § 4. Attach to the orbitSSS(w) the complex ofsheaves

C(w)R = j (w)!R [dw].Remark.From the discussion in [CC] (§ 3) it follows that the complexof sheavesC(w)C corresponds to a generalized Verma module via theRiemann–Hilbert correspondence and localization and thus is in PervKKK∩PPP(XXX,{SSS(w)},C).

We shall have to lift the BGG resolution toKKK ×KKK∩PPP

XXX. So we define the

inclusiong(w) : KKK ×KKK∩PPP

SSS(w)→ KKK ×KKK∩PPP

XXX. We set

C(w)R = g(w)!R[n+ dw],R a constant sheaf onKKK ×

KKK∩PPPSSS(w).

The maps in what will be the sheaf version of theBGG resolutionarise from two fibrations involving the geometry of flag manifolds. Recallthat every simple reflections in S, determines a parabolic subgroupPPP(s)containingBBB and a generalized flag manifoldXXX(s) = GGG/PPP(s).


The first map comes from the fibration

P1 −→ XXXπs−→ XXX(s)

or analogously from

KKK ×KKK∩PPP(s)∩LLL

PPP(s)/BBB −→ KKK ×KKK∩LLL

XXXπs−→ KKK ×

KKK∩LLLXXX(s).

Define a set-map R(s): XXX→ XXX or R(s): KKK ×KKK∩LLL

XXX→ KKK ×KKK∩LLL

XXX by

R(s)= π−1s ◦ πs. Then one has the diagram

XXXR(s)−→ XXX

πs↘ ↙ πs

XXX(s)

or the diagram

KKK ×KKK∩LLL

XXXR(s)−→ KKK ×

KKK∩LLLXXX

πs↘ ↙ πs

KKK ×KKK∩LLL

XXX(s)

(5.2)

The mapπs gives functors on D(XXX, {XXXk}, R) or on D(KKK ×KKK∩LLL

XXX, {KKK ×KKK∩LLL

XXXk}, R)

by definingRc(s)F = π !sπs!(F )[1],

R(s)F = π∗sπs!(F )[1].(5.3)

Now suppose thatw andsw are elements ofWPPP with l(sw) = l(w)+1, andlet SSS(w) andSSS(sw) be the correspondingPPP-orbits in XXX. ThenR(s)(SSS(w))is a smooth manifold containingSSS(sw), and which fibers overSSS(w) withfiberP1

P1→ R(s)(SSS(w))→ SSS(w).

Then we obtain as a refinement of (4.2)

0→ j (sw)!RS(sw)→ RR(s)(S(w))→ j (w)!RS(w)→ 0(5.4)

an exact sequence of sheaves. (It is useful to compare this to (2.4).) Asdescribed before, using a mapping cone construction we obtain a map inthe derived category

j (w)!RS(w)→ j (sw)!RS(sw)[1].


Translating everything bydw we obtain our first map

C(w)R→ C(sw)R.

In a similar way forKKK ×KKK∩LLL

XXX, but translating byn+ dw, we obtain

C(w)R→ C(sw)R.

In particular, overC this gives a map between perverse sheavesC(w)C →C(sw)C which is, in the language of generalized Verma modules, a standardmap. These are, however, not all of the standard maps which appear in theBGG resolution in [Le].

The second map in the sheaf version of theBGG-resolution uses theproduct manifoldXXX × XXX consisting of all pairs of Borel subgroups ofGGG.We have the diagram

XXX × XXX

p1↙ ↘ p2

XXX XXX

(5.5)

Here p1 (resp. p2) is the projection on the first (resp. second) factor. Foreachs in S, we have a smooth submanifoldXXXs of XXX × XXX consisting ofall pairs(BBB,BBB′) of Borel subgroups in relative positions. Thus,BBB andBBB′generate a parabolic subgroupPPP(s). Restricting the diagram (5.5) to thesubmanifoldXXXs and denoting the projections byp1(s) and p2(s) we obtainvia this correspondence the diagram

XXXs

p1(s)↙ ↘ p2(s)

XXXT(s)−→ XXX

or the diagram

KKK ×KKK∩LLL

XXXs

p1(s)↙ ↘ p2(s)

KKK ×KKK∩LLL

XXXT(s)−→ KKK ×

KKK∩LLLXXX

That is, we define a set-mapT(s) : XXX→ XXX or T(s) : KKK ×KKK∩LLL

XXX→ KKK ×KKK∩LLL

XXX

by T(s) = p2(s) ◦ p1(s)−1.


Then we obtain functors on D(XXX, {XXXk}, R) or D(KKK ×KKK∩LLL

XXX, {KKK ×KKK∩LLL

XXXk}, R)

byTc(s)F = p2(s)!p1(s)

∗F [1],

T(s)F = p2(s)∗p1(s)∗F [−1].(5.6)

We can now describe a map betweenC(xy)R andC(xsy)R, wherexy ∈WPPP, xsy∈ WPPP, s is simple, andx increases the length of bothy andsy. Letx = s1 . . . sj−1 be a reduced expression ofx. Suppose we have a standardmap f : C(y)R→ C(sy)R as already described. If we apply the functors ofthe formTc(si ) successively to the mapf we obtain a map

Tc(s1). . .Tc(sj−1) f : Tc(s1). . .Tc(sj−1)C(y)R→Tc(s1). . .Tc(sj−1)C(sy)R,

that is, a non-zero map

C(xy)R→ C(xsy)R.

Similarly we obtain a non-zero map

C(xy)R→ C(xsy)R.

To see that in the case of R =C this corresponds to a standard map in [Le],it suffices to consider the case in whichPPP = BBB, for a standard map betweengeneralized Verma modules is obtained from the (unique) map betweencorresponding Verma modules by passing to the quotient. In the case ofPPP = BBB, the map above is clearly non-zero, thus is the unique non-zero mapbetween the corresponding Verma modules via the dictionary of [BB].

We refer to either of the two types of maps thus constructed as standardmaps.

Remark.It will be important to note a simple relation betweenTc(s)andT(s).Namely, forw, s with l(sw) = l(w) + 1, we have thatT(s)Tc(s)C(w)R

∼=C(w)R and T(s)Tc(s)C(w)R

∼= C(w)R. The proof of this fact reduces toa P1 calculation, corresponding to the case ofsl(2). In the case ofP1

the stalks ofT(s)Tc(s)C(w)R are obtained by computing cohomology overCx = P1 \ { x} of a constant sheaf in degree−1 onC extended by zero,and then shifting by[−1]. The cohomology computation without the shiftsgivesH∗(Cx,∞; R) = 0 if x 6= ∞, andH∗(C; R) if x = ∞. Thus, as theshifts [1], in the original perverse sheaf, and[−1] cancel each other, onehas a skyscraper sheaf in degree zero supported at∞.

Orbit parameters.

The mapsR(s) andT(s) have a useful geometric description on orbits.For PPP-orbits letw be in WPPP and take any reduced expression ofw =s1 . . . sn. Consider

R(s1) . . . R(sn)({SSS(e)}),


SSS(e) = PPP/BBB the PPP-orbit of smallest dimension. ThePPP-orbit, SSS(w), associ-ated tow is the largestPPP-orbit inside this union ofPPP-orbits.

If OOO is a KKK -orbit , thenR(s)(OOO) (s ∈ S) is a union ofKKK -orbits. Thestructure of these orbits definesT(s)(OOO). If R(s)(OOO) consists of two orbitsboth of which intersect the fibers ofπs in connected sets, thenT(s)(OOO) is theKKK -orbit in R(s)(OOO) − {OOO}. If R(s)(OOO) contains either three orbits, or twoorbits one of which intersects the fibers ofπs in two points, thenT(s)OOO = OOO.This extends tow in WPPP. Take any reduced expressionw = s1 . . . sn andOOO a KKK -orbit containing a point corresponding to a Borel subalgebra con-tainingh. ThenT(w)(OOO) = T(s1) . . . T(sn)(OOO) is well defined and dependsonly onw.

Recall thatOOOo = KKK/KKK ∩ BBB is the openKKK -orbit in XXX. We setO(w) =T(w)(OOOo). The orbitO(w) is theKKK -orbit that is dense in the support of theperverse sheaf associated toJ(ξ(w)) by the functorP . Notice that it mayhappen thatw 6= w1, whileO(w) = O(w1). For example, in [Ma] one findsthe characterization ofWPPP

K (i.e. WP) as thosew such thatO(w) = {OOOo}.GeometricBGG resolution forZ.

(5.7) Definition. A geometric version overC of theBGG resolution onXXXis a cellular resolution relative to{XXXk} of a constant sheaf onXXX by (shifted)perverse sheaves with a trivialKKK ∩ PPP-action

· · · −→⊕

l(w)=kw∈WPPP

C(w)C [−n] →⊕

l(w′)=k+1w′∈WPPP

C(w′)C [−n] −→ · · · .

The mapC(w)C → C(w′)C is given by sendingv → s(w,w′) f(w,w′)vwith f(w,w′) a standard map betweenC(w)C andC(w′)C described above.A geometric version overRof theBGGresolution onKKK ×

KKK∩PPPXXX is a cellular

resolution

· · · −→⊕

l(w)=kw∈WPPP

C(w)R[−n] −→⊕

l(w′)=k+1w′∈WPPP

C(w′)R[−n] −→ · · · .(5.8)

SetC

kR =

⊕l(w)=kw∈WPPP

C(w)R[−n]

and denote the resulting diagram byC•R.

We consider the mapp: KKK×XXX→ XXX given by projection onto the secondfactor. IfF is in PervKKK∩PPP(XXX, {SSS(w)},C), p∗F [dimC KKK ] is in PervKKK∩PPP(KKK ×XXX, {KKK × SSS(w)},C). In the latter we also useKKK ∩ PPP to refer to the diagonal


action onKKK ×XXX. Let P be the functorP = p∗[dimC KKK ], p∗ composed withthe dimension shift by dimC KKK , so that

P : D(XXX, {SSS(w)}, R)→ D(KKK × XXX, {KKK × SSS(w)}, R).

We denote byq the quotient mapq : KKK × XXX→ KKK ×KKK∩PPP

XXX. The functor

Q = q∗[dimC KKK ∩ PPP] induces an equivalence of categories between

Per(KKK ×KKK∩PPP

XXX, {KKK ×KKK∩PPP

SSS(w)},C)and PervKKK∩PPP(KKK × XXX, {KKK × SSS(w)},C)

([Bi], Lemma 4.2). Denote byQ−1 the inverse of the category equivalenceQ so that

Q−1 : PervKKK∩PPP(KKK×XXX, {KKK×SSS(w)},C)→ Per(KKK ×KKK∩PPP

XXX, {KKK ×KKK∩PPP

SSS(w)},C).

Remark.If we useC(w)Z we obtain a cellular resolution of the constantsheafZ on KKK ×

KKK∩PPPXXX. Applying base change to this givesC(w)C , a cellular

resolution of the constant sheaf onKKK ×KKK∩PPP

XXX by perverse sheaves (all shifted

by −n). Moreover if we apply the functorQ−1 to the cellular resolutionin (5.7) we obtain (up to shift) a resolution ofKKK × XXX by elements inPervKKK∩PPP(KKK × XXX, {KKK × SSS(w)},C).(5.9) Proposition. The diagram (5.8) of maps is a2−act∗-acyclic resolutionoverZ. If in addition we assume thatHq(Yp,Yp−1;Z) = 0 if q 6= p, andH p(Yp,Yp−1;Z) is torsion free, then the diagram (5.8) of maps is also anact∗-acyclic resolution overZ.

Proof. The 2-acyclicity or acyclicity is a consequence of (4.16) or of theadditional assumption in the case of the proof of b). We show that (5.8) infact gives a cellular resolution of the constant sheaf. First we remark thatZis projective asZ module, so that Ext∗Z(Z,Z) is just HomZ(Z,Z). Next, inthe derived category we have that

Hom(C(xy)Z, C(xsy)Z) = Ext∗Z(Z,Z) = HomZ(Z,Z),

wherex, yare in the Weyl group ofGGG, s is simple, andl(xsy) = l(xy)+1.Theonly possible difference between the maps in the definition of a cellularresolution and (5.8) consists of signs. However any other choice of signssimply gives another family of signs as in (1.3). This results from thecondition that the composition of two differentials must be zero. ut(5.10) Definition.We define a relation→→ in WP and a graphGrep

∗ . Toeach element ofWP associate a vertex. Two verticesw andw′ with l(w′) =l(w)+ 1 are connected with an oriented edge→→ if and only if the map


f : J(ξ(w))→ J(ξ(w′)) contains an asymptotically large representation inits image.

The following is a new application of Takeuchi’s theorem.

(5.11) Theorem. The differentials of the chain complex

Cell∗({Xp};Z) =· · ·→Hp(Xp, Xp−1;Z)→Hp−1(Xp−1, Xp−2;Z)→ · · ·are matrices with entriesa(x, y) with x,y in WP satisfying

a(x, y) 6= 0 if and only ifx→→ y.

Proof.We use (4.18) and 5.9. utThe functorL.

Following Bernstein, we shall define a functor that will correspond tothe functorL used in § 1. In fact we defineL andpL, the perverse part ofL.This will allow us to formulate geometric versions of bothI ∗(BGG) andJ∗(BGG).

We use the functorsQ−1 and P. Let D(XXX, {Oi },C) be the derivedcategory with{Oi } the KKK -orbits, and define the functorL by L = act∗ ◦Q−1 ◦ P, so that

L : D(XXX, {SSS(w)},C)→ D(XXX, {Oi },C).If F is in D(XXX, {SSS(w)},C), define pL(F ) by pL (F ) = pH0act∗ Q−1

P(F ), so that

pL : PervKKK∩PPP(XXX, {SSS(w)},C)→ PervKKK (XXX, {Oi },C).

(5.12) Theorem. (Bernstein)If M is in OPPPρ , then

P (L(M)) = pL(P (M)).

Remark.The functorP is used rather thanP shifted by[dimXXX] becausethe first functor is covariant and the second contravariant. For a proof of theTheorem see [Bi].

We may now give the proof of Lemma (4.13) IV. For convenience werecall the statement.

(4.13) Lemma. IV) H j (Yp,Yp−1;Z) free= 0 if j 6= p.

Proof. The proof of IV can be obtained in a roundabout way by first in-terpreting H∗(Yp,Yp−1;C) as in the first part of the proof. Thus eachH∗(Yp,Yp−1;C) is obtainable by applying act∗ to a constant sheaf extendedby zero onKKK ×

KKK∩PPPSSS for a suitableS, and then restricting to a pointx in the


openKKK orbit. Under the equivalence of categories described in [BB] or alsoin 5.1 of section 5, the complex of sheaves, before restriction to a pointx,corresponds to an induced representation. This remark uses [5.12] which isproved in [Bi]. Thus one can interpret first as the perverse sheaf version ofLepowsky’s generalization of the BGG resolution described in (1.4), andalso conclude that when the functorL is applied to (1.4) one obtains a chaincomplex of induced representations dual to (1.9).

Therefore, in view of this dictionary between representation theory andthe theory of perverse sheaves, we see that we are just considering therestriction of a certain perverse sheaf to a pointx. Such a perverse sheafrestricted to the openKKK orbit lives in exactly one degree. This shows thatH j (Yp.Yp−1;C) = 0 if j 6= p. This implies, using the universal coefficienttheorem for cohomology, thatH j (Yp,Yp−1;Z) free= 0 if j 6= p. ut

6. n-homology – sheaf version

In § 3n-homology provided a complex of finite dimensional spaces quasi-isomorphic to the BGG complex. In this section we use the geometricversion ofn-homology to determine the standard maps in the geometricversion of the BGG resolution. We begin by recalling the characterizationof the derived sheaves ofP (M)(see [VoIII] ).

(6.1) Theorem. Let M be inHHHCCCρ or OPPPρ . For anyx ∈ XXX let BBBx denote the

isotropy group ofx andbx = hx ⊕ nx its Lie algebra. Then

Hq(P (M))x = HomC (Hq(nx;M)0,C) ,i.e. the dual of the0 weight space innx-homology.

The following example is useful to keep in mind during the subsequentcalculations of the standard maps.

(6.2) Example.Let G be SL(2,R) and K = SO(2). ThenGGG = SL(2,C),KKK = SO(2,C) andXXX ∼= P1 = C ∪ {∞}. TheKKK-orbits in XXX correspond toC − {0}, {0}, and{∞}. Let j : C − {0} → XXX be the inclusion of the openKKK -orbit. Define a presheaf ofZ-modules,LZ, onC − {0} by assigning toany open setU of C − {0} the integral multiples of a branch of

√z on U.

ThenLZ is a sheaf and the sheafL = LZ⊗ZC is aKKK -equivariant sheaf thatdoes not extend toP1. Moreover, if M0 ∈ HHHCCCρ is an irreducible principalseries representation forSL(2,R), thenP (M0) ∼= j!L ∼= j∗L (see [VoIII],[LV]).

There is a standard map between Verma modules that induces an iso-morphism onM0. We describe this map first in geometric terms as a map be-tween complexes of sheaves overZ. Tensoring withCgives the map betweenperverse sheaves. LetW = {s,e} be the Weyl group. Using base change onesees that the cone of the map act∗C(e)Z→ act∗C(s)Z is R(s)act∗C(e)Z.This is a complex with cohomologyH∗(P1; j∗LZ) ∼= H∗(C − {0};LZ).


Then one obtainsH0(C − {0};LZ) = 0 andH1(C − {0};LZ) = Z/2Z.Consequently, the map betwen the two complexes of sheaves is non-zeroon the level of stalks on the largestKKK -orbit in the support. Now restrictionto P1 = π−1

s πs(x), x in the openKKK -orbit, gives exactly the map

j∗L→ j!L.

Therefore, on stalks on the openKKK orbit this map is multiplication by 2.utNilpotent Lie Algebra Homology.

We shall examine a typical map in the BGG complex arising from (5.4).Fix w ∈ WPPP and recall from § 5 theKKK -orbitO(w). Fors ∈ S let b1 andb2be two Borel subalgebras in relative positions. Suppose thatb1 contains theLie algebrah and is inO(w). We may assume thatb1 = w−1b and thereforeb2 = w−1sb. If p(s) = l(s)⊕ u(s) is the parabolic subalgebra generated by〈b1, b2〉, then[l(s), l(s)] ∼= sl(2,C) with [l(s), l(s)] coming fromG or notaccording to whether or not the root associated tos is real or complex. Ofcoursep(s) depends onw but no confusion should result from omitting thisfrom the notation. Setq = codimR(s)O(w).

We are concerned only with modules having trivial infinitesimal char-acter. So for a representation (or formal character) ofG we will denoteby P0 the projection onto that part having trivial generalized infinitesimalcharacter. Similarly, for a Levi-factor we havePL

0 .

(6.3) Lemma. Letp(s) = l(s)⊕ u(s) be a parabolic subalgebra generatedby two Borel subalgebras〈b1, b2〉 in relative positions (s ∈ S). Supposethat b1 contains the Lie algebrah and is inO(w) for somew ∈ WPPP. ThenPL

0 Hk(u(s); J(ξ(w))) = 0 unlessk = q = codimR(s)O(w).

Proof. This fact is well-known but we include a proof. This argumentis analogous to an argument used in [VoIII]. AsP (J(ξ(w)))[dimXXX] isa perverse sheaf andR(s)O(w) is open in its support, its restriction toR(s)O(w) is also perverse sheaf shifted by[−q]. Consider the fibrationassociated tos,

P1 −→ XXXπs−→ XXX(s).

Further restriction fromR(s)O(w) to P1 will give rise to a perverse sheafshifted also by[−q].

We have that for any Harish–Chandra moduleJ and a projective reso-lution P∗ of J by Aρ modules:

HomAρ

(P∗,Homp

(U(g),OP1

))= RHomU(l(s))

(P∗/u(s)P∗,OP1

)Therefore the restriction ofP (J) to any fiberP1 of the fibration above,may be interpreted as the complex of sheaves onP1 associated to the com-plex of l(s) modules(P∗/u(s)P∗) by the functorP L , defined analogously


to P and applied to modules for the Levi-factor ofp(s). The cohomo-logy of (P∗/u(s)P∗) is H∗(u(s); J) and if J = J(ξ(w)), we obtain thatPL

0 Hk(u(s); J(ξ(w))) = 0 unlessk = q = codimR(s)O(w) because onlythe summandPL

0 H∗(u(s); J) is detected with the constructionP L .Using the spectral sequenceHl(CXα; (Hk(u(s); X)) ⇒ Hl+k(CXα ⊕

u(s); X), whereXα is a root vector inw−1n and in the Levi-factor, togetherwith the condition on the stalks atx in O(w) of the derived sheaves and(6.1), it follows that the restriction ofP (J(ξ(w))) to the fiber containingb1is quasi-isomorphic toP L(PL

0 Hq(u(s); J(ξ(w)))). ut(6.4) Lemma. Letp(s) = l(s)⊕ u(s) be a parabolic subalgebra generatedby two Borel subalgebras〈b1, b2〉 in relative positions (s ∈ S). Supposethat thebi are in R(s)(O(w)). Then

a) for any Harish–Chandra moduleX and anyk,

PL0 Hk(u(s);φsψsX) ∼= φL

sψLs P

L0 Hk(u(s); X);

b) the exact sequence0 → J(ξ(sw)) → φsψsJ(ξ(w)) → J(ξ(w)) → 0induces a short exact sequence:

0→ PL0 Hq(u; J(ξ(sw)))→ φL

sψLs P

L0 Hq(u(s); J(ξ(w)))

→ PL0 Hq(u(s); J(ξ(w)))→ 0 .

Consequently, if the pair(J(ξ(w)), J(ξ(sw))) is in relative positionsthen the pair(PL

0 Hq(u(s); J(ξ(w))),PL0 Hq(u(s); J(ξ(sw)))) is in relative

positions.

Proof.See Theorem 4.2b [VoII] together with (6.3), thatPL0 Hk(u(s); J(ξ(t)))

vanishes except forq = codimR(s)O(w), t = sw,w. utFor p(s) = l(s) ⊕ u(s) with s associated to the simple real rootα,

the Levi-factorl(s) is the complexified Lie algebra of a subgroup ofGcontaining a copy ofSL(2,R) or Ad SL(2,R). As described in § 2 ifG isa group with ample involutions there is a map,σα, on g, implemented bya certain elementz ∈ GGG, that extends the involutionσ . The map preservesl(s) andu(s). Also, if (π, X) is any module forsl(2,C), then twisting withσgives another module (πσ, Xσ ) on whichZ in sl(2,C) acts byπ ◦σ(Z). Wealso form the twistedg-module(πσα, Xσα) for anygmoduleX. We remarkthat while the real rootα is simple here, the order need not be aτ-order(recall from § 2 that we use the notation of [Wa]). This is the reason that weconstructedσα for all real roots.

(6.5) Lemma. Let s ∈ S be associated to the simple real rootα, and letp(s) = l(s) ⊕ u(s) with [l(s), l(s)] ∼= sl(2,C). Suppose thatα has a ampleinvolutionσα. For anyr , we have an equivalence asl(s)-modules

Hr (u(s); Xσα) ∼= Hr (u(s); X)σ .


Proof. The chain complex that computes the nilpotent homologyHr (u(s);Xσα) (resp.Hr (u(s); X)) is of the form

· · · → ∧ku(s)⊗ Xσα → ∧k−1u(s)⊗ Xσα → · · ·Here elementsZ in u(s) act on Xσα by π ◦ σα(Z) (resp. onX by π(Z)).Recalling thatσα is an isomorphism onu(s), it is easy to check that the map,onu(s), Z→ σα(Z) induces an isomorphism of complexes

∧∗u(s)⊗ Xσα ∼= ∧∗u(s)⊗ X,

and hence an isomorphism of vector spacesHr (u(s), Xσα) ∼= Hr (u(s), X).This map of complexes, induced byZ→ σα(Z), intertwines thel(s)-actionon∧∗u(s)⊗Xσα with thel(s)-action on∧∗u(s)⊗X then twisted byσ . Thusasl(s)-modules,Hr (u(s), Xσα) ∼= Hr (u(s), X)σ . ut(6.6) Corollary. Let s ∈ Sbe associated to the simple real rootα, and letp(s) = l(s) ⊕ u(s) with [l(s), l(s)] ∼= sl(2,C). Let (π, X) = IndG

P(τ ⊗ eν).

For anyr , we have an equivalence as[l(s), l(s)]-modules

Hr (u(s); X)σ ∼= Hr (u(s); Xσα).

Proof.This follows from (6.5) and the remarks in § 2 following (2.9).utTheKKK -orbit structure ofR(s)O(w) has few possibilities. One of these is

whenR(s)O(w) contains precisely threeKKK -orbits. The next several Lemmasconsider this case in detail.

(6.7) Lemma. Let s ∈ S be associated to the simple real rootα and letO(w) be aKKK -orbit containing two Borel subalgebrasb1 andb2 in relativepositions. Suppose thatR(s)(O(w)) is the union of threeKKK -orbits,

R(s)(O(w)) = OOO+ ∪ OOO− ∪ OOO,

with dimOOO+ = dimOOO− = dimOOO − 1, and OOO = O(w). Let (OOO+,L+),(OOO−,L−) be two pairs inD and suppose there is aKKK-equivariant localsystemL on OOO, which extends toR(s)(O(w)) and whose restriction to theorbit OOO+, resp.OOO−, is L+, resp.L−. Then, withq as in (6.3) we have anisomorphism of[l(s), l(s)] modules:

PL0 Hq(u(s);π(OOO+,L+))σ ∼= PL

0 Hq(u(s);π(OOO−,L−)).(6.8)

Proof.We claim that the[l(s), l(s)]modules in the statement are irreduciblehighest weight modules isomorphic to the modulesD+ and D− describedin § 2 . Asα is a real root there is a Lie isomorphismΦα from sl(2,R)to [l(s)0, l(s)0] and an involutionσ that acts on[l(s), l(s)]-modules. Theinvolution σ exchanges the root vectors with respect to a compact Cartanh∗ of sl(2,R) and thus exchangesD+ andD−.


To prove the claim consider the fibration associated tos,

P1 −→ XXXπs−→ XXX(s).

Let R (∼= P1) be the fiber in the flag manifold containingb1. The modulesπ(OOO±,L±) whose homology we are computing, correspond (see [BB] and[VoIII]) to certain perverse sheaves via (5.1). Each intersectionOOO± ∩ Rconsists of one point, and the two perverse sheaves restricted toRgive a pairof skyscraper sheaves onR with support this point. The data(OOO±,L±)simply give information to reconstruct these perverse sheaves. In thesl2theory, these skyscraper sheaves are attached toD+ andD−. From here onecan easily deduce the[l(s), l(s)] module structure ofPL

0 Hq(u(s), . . . ). ut(6.9) Lemma. Letp(s) = l(s)⊕ u(s) be a parabolic subalgebra generatedby two Borel subalgebras〈b1, b2〉 in relative positions (s ∈ S). Supposethat thebi are in R(s)(O(w)). Assume thatR(s)(O(w)) consists of threeorbits as in (6.7). Then as[l(s), l(s)]-modules,

PL0 Hq(u(s), J(ξ(w)))σ ∼= PL

0 Hq(u(s), J(ξ(w))),

PL0 Hq(u(s), J(ξ(sw)))σ ∼= PL

0 Hq(u(s), J(ξ(sw)))

Proof.This follows from (6.6) and (2.9) . utA standard map in the BGG complex will induce anl(s)-module map

on the stalks of the derived sheaves. Therefore we shall compute thel(s)-module structure ofPL

0 Hq(u(s); J(ξ(w))). This is most conveniently doneby means of its formal character. We review this known material on formalcharacters briefly.

For any parabolic subalgebrap = l⊕ u, u the nilradical, and ag-mod-ule X, the vector spacesHk(u; X) arel-modules. Assume thatX is a Harish–Chandra module. Ifp is a Borel subalgebra andl is the complexification ofa realθ-stable Cartan subalgebral0 of g0, then the global character ofXrestricted to a certain open region inl0 is the quotient of the alternating sumof the characters of thel-modulesHk(u; X) by a Weyl denominator. This isessentially the statement known as Osborne’s conjecture and follows fromthe results in [HS] together with Theorem 8.1 [VoII]. Hence forb = h⊕ nany Borel subalgebra containingh, one can define the formal character,ch X, of X by

ch X = Σ(−1)kchHk(n; X)/Σ(−1)kch∧k n .

Since the Levi-factor ish, chHk(n; X) is the trace of the(h, T )-action onthe (finite dimensional) homology group. The character chX turns out tobe the character of a categoryO module forg (with respect to abx wherexis in the openKKK -orbit).


If X has trivial infinitesimal character, the numerator in chX (ignoringtheT-action) is an integral combination of terms of the formeρ(n)−σρ(n), sothat one has

ch X = Σawe−wρ(n)+ρ(n)/Σ(−1)kch∧kn.

As long as the Cartan subalgebra remainsh, the formal character chX isindependent of the choice of nilradical so by varying the Borel one obtainsvarious identities. For example, if a different choice of nilradical were made,sayσ n, the formal character becomes

ch X = Σa′we−wρ(σ n)+ρ(σ n)/Σ(−1)kch∧k σ n,

where a′w = (−1)l(σ)awσ . In particular, if wσ is the identity e, a′t =(−1)l(σ)ae, t = σ−1. Similarly if b corresponds to a point inO(w) sothat n = w−1n, thena′t = (−1)l(w)ae, t = w−1. While if two Borel subal-gebras containingh are in relative positions, for a simple reflections in S,the two coefficientsa′t, a′ts are determined byae andas.

We shall specialize these considerations to a moduleX of type J(ξ(w))and a parabolicp(s) = l(s) ⊕ u(s). We view the two Borel subalgebras astwo points inR(s)O(w). Here the Levi-factor action onPL

0 Hq(u(s); J(ξ(w))is a Harish–Chandra module ifl(s) is generated by two opposite real roots,or a categoryO module forsl(2,C) if [l(s), l(s)] is generated by complexroots. Using the spectral sequenceHl(CXα; (Hk(u(s); X))⇒ Hl+k(CXα⊕u(s); X), whereXα is a root vector inw−1n and in the Levi-factor, we obtainonh ∩ [l(s), l(s)],

chPL0 Hq(u(s), J(ξ(w))) = (a+ beα)/(1− eα)(6.10)

So(−1)l(w)a is the coefficient ofe−tρ+ρ and(−1)l(w)b is the coefficient ofe−tsρ+ρ in ch J(ξ(w)), t = w−1. As the characters of the modulesJ(ξ(w))are known, we can obtain character formulas for the modules that occur inPL

0 Hq(u(s); J(w)). In this way one obtains

(6.11) Lemma.Letp(s) = l(s)⊕u(s) be a parabolic subalgebra generatedby two Borel subalgebras〈b1, b2〉 in relative positions (s ∈ S). Supposethat thebi are in R(s)(O(w)), w ∈ WPPP. Let thechPL

0 Hq(u(s), J(w)) on[l(s), l(s)] ∩ h be(a+ beα)/(1− eα).

a) If l(s) contains a real root vector,a= 1, b= 1.b) If [l(s), l(s)] is generated by two opposite complex roots andO(w) is

open inR(s)(O(w)), thena= 0, b= 1.

c) If [l(s), l(s)] is generated by two opposite complex roots andO(sw) isopen inR(s)(O(w)), thena= 1, b= 0.

Using the technique of reduction tosl2 and the results in § 2 we shallobtain more detailed information about the modulesPL

0 Hq(u(s), J(ξ(w))).


Recall from (6.2) thatM0 denotes the irreducible principal series represen-tation with trivial infinitesimal character. Also we recall from § 2 thatMdenotes the principal series representation having trivial infinitesimal char-acter and having the trivial module as submodule, as well as the functorsendingM to M.

(6.12) Lemma.Letp(s) = l(s)⊕u(s) be a parabolic subalgebra generatedby two Borel subalgebras〈b1, b2〉 in relative positions (s ∈ S). Supposethat thebi are in R(s)(O(w)), w ∈ WPPP. Suppose thatR(s)(O(w)) consistsof three orbits as in (6.7). Also assume thatsw ∈ WPPP andl(sw) = l(w)+1.Then there is a moduleM1 such that

PL0 Hq(u(s), J(ξ(w))) ∼= M1,

PL0 Hq(u(s), J(ξ(sw))) ∼= M1 .

The[l(s), l(s)]-moduleM1 is isomorphic either toM0 or to M.

Proof.The character of either of these modules is the same as the character ofM or the character ofM0. For M0 there is nothing to prove sinceM0

∼= M0.To handle the other case, we observe from (6.4)b) together with (6.9) thatthis module is a Harish–Chandra moduleZ for [l(s), l(s)] satisfying

a) φLsψ

Ls (Z)→ Z is surjective.

b) Zσ ∼= Z.c) Z has composition factorsD+, D−,C.

Using Lemma (2.12), we obtain the desired result. utWe summarize the situation in the next Proposition. For the statement,

we recall that[l(s), l(s)] is ansl(2,C) and from § 2 the modules denotedthereM0, M, M′, M, andM′.

(6.13) Proposition. Let p(s) = l(s) ⊕ u(s) be a parabolic subalgebragenerated by two Borel subalgebrasb1, b2 in relative positions (s ∈ S).Assume thebi containh, correspond to positive root systems∆+1 , ∆+2 , andare in R(s)O(w) w ∈ WPPP. Also, suppose thatsw ∈ WPPP and l(sw) =l(w)+ 1. Then withq = codimR(s)O(w)

a) PL0 Hk(u(s), J(ξ(t))) = 0 for k 6= q and t = w or sw.

b) PL0 Hq(u(s), J(ξ(w))) is an indecomposable[l(s), l(s)]-module, sayF .

If F is irreducibleF ∼= M0 or it is isomorphic to an irreducible Vermamodule. IfF is reducible,F contains the trivial module as a submodulewith multiplicity one, andF is isomorphic to eitherM or M′.

c) PL0 Hq(u(s), J(ξ(sw))) is an indecomposable[l(s), l(s)]-module, sayF .

If F is irreducible,F ∼= M0, or it is isomorphic to an irreducible Vermamodule. IfF is reducible,F contains the trivial module exactly once asa quotient, andF is isomorphic to eitherM or M′.


d) Suppose that the mapJ(ξ(w)) → J(ξ(sw)) induces a non-zero mapPL

0 Hq(u(s), J(w)) → PL0 Hq(u(s), J(ξ(sw))), then this map is an iso-

morphism of [l(s), l(s)]-modules provided the induced mapHq(n,J(ξ(w)))0→ Hq(n, J(ξ(sw)))0 is non-zero for all but a finite number ofLie algebrasn, which are a nilradical of a Borel subalgebra containedin p(s).

e) Letσ = s1 . . . sj−1 and assume it increases the length ofw and sw byj − 1. Let p(σ) = l(σ) + u(σ) denote the parabolic containingh andgenerated by two Borel subalgebras with positive root systemsσ∆+1 ,σ∆+2 . Suppose thatJ(ξ(w)) → J(ξ(sw)) induces a mapPL

0 Hq(u(σ),

J(ξ(w)))→ PL0 Hq(u(σ), J(ξ(sw))) which is an isomorphism. Then

PL0 H∗(u(σ), J(ξ(σw))) ∼= PL

0 H∗(u(σ), J(ξ(σsw))).

Proof. There are several cases depending on which kind of roots generate[l(s), l(s)]. The character ofPL

0 Hq(u(s), J(ξ(t))) is given by Lemma (6.11)on the level of the Cartan subalgebra of[l(s), l(s)]. There are several possi-bilities for the group. If[l(s), l(s)] is generated by two opposite real roots,the action of the Levi factor will give rise to a Harish–Chandra module forSL(2,R) or SL(2,R)±, or an adjoint group. Here we obtain an irreducibleprincipal series orM or M′ by using (6.12) and (2.12). If[l(s), l(s)] is gener-ated by complex roots, using (2.13) we obtain a dual of a Verma module oran irreducible Verma module. Then d) follows because for all the modulesV listed, dimHom(V,V) = 1. For e) if the length ofσ is zero this is d).Otherwise proceeding by induction on the length ofσ , we use (6.11) or(6.12)a), d) above, and the construction in (2.7). ut(6.14) Lemma. AssumeG is a group with ample involutions. Letw, w′ inWPPP be such thatO(w) = O(w′). Assume thats ∈ S increases the length ofw andw′ and that there is a mapf : J(ξ(w))→ J(ξ(w′)). Then f inducesan isomorphism on the0-weight space inn′-homology

Hq(n′, J(ξ(w)))0→ Hq(n

′, J(ξ(w′)))0

for n′ a nilradical of a Borel subalgebra corresponding to a point inO(w),if and only if f(s) induces an isomorphism

Hq(n′′, J(sw))0→ Hq(n

′′, J(ξ(sw′)))0

for anyn′′ corresponding to a point inO(sw).

Proof. It suffices to prove that iff induces an isomorphism in cohomo-logy, so doesf(s). We have shown that ifp(s) = l(s) ⊕ u(s) is gener-ated by two opposite Borel subalgebras inR(s)O(w) containingh, thenPL

0 Hq(u(s), J(ξ(w))) is either irreducible or contains as unique irreducible


submodule the trivial module. The mapf then induces an isomorphism asin (6.13e)

f : PL0 Hq(u(s), J(ξ(w)))→ PL

0 Hq(u(s), J(ξ(w′))).

If l(s) is generated by two opposite real roots, then the two modules involvedcan be identified withM or M′ or an irreducible principal series moduleM0. A map between them, which is non-zero has to be an isomorphism.Since we obtain two pairs of modules(M1, M1), (M2, M2) in relative pos-ition s, the isomorphismf : M1 → M2, inducesf(s) : M1 → M2 whichmust also be an isomorphism. If we have an irreducible principal seriesM0,it is done similarly. Using the spectral sequenceHq(CXα, Hk(u(s), X))⇒Hq+k(CXα + u(s), X), the result follows.

If l(s) is generated by complex roots, we have a map between twoirreducible Verma modules or two duals of Verma modules each containingthe trivial representation. Again a non-zero map has to be an isomorphismso the result follows as before. utRemark.Letw ∈ WPPP and letα be a real root. Recall thatn+ is the nilradicalof pmin. Setmw,α = 〈α, ξ(w)ρ + ρ(n+)〉. This notation,mw,α, is simply thedefinition of “n” in [VoI] Theorem 4.12 for the principal seriesJ(ξ(w)) andmodified to reflect that we do not use normalized induction whereas Vogandoes. Also we refer to [VoI] p.82 for the definition of the signεα. It wouldbe interesting to give a geometric formula formw,α andεα but we will notneed one here.

(6.15) Lemma. Assume thatG is a group with ample involutions. Letw, sw ∈ WPPP be such thatO(w) = O(sw). Assume thatl(sw) = l(w)+ 1.A map f : J(ξ(w)) → J(ξ(sw)) induces an isomorphism on the0-weightspace inn′-homology

Hq(n′, J(ξ(w)))0→ Hq(n

′, J(ξ(sw)))0

for n′ a nilradical of a Borel subalgebra corresponding to a point inO(w),if and only if

a) w−1α is a real root,

b) there is a Lie group homomorphismΦ : SL(2,R)→ G, whose imagehas Lie algebra generated by the root vectors corresponding tow−1αand−w−1α for whichχ(ξ(w)(ΦDiag(−1,−1)) = −(−1)mw,αεα.

Proof. The proof is analogous to the proof of the previous result. Note thatthe condition thatO(w) = O(sw) implies that the parabolic subalgebrap(s)has its Levi-factor generated by two opposite real roots. We consider a map

PL0 Hq(u(s), J(ξ(w)))→ PL

0 Hq(u(s), J(ξ(sw))) .


The left side can be only the principal series moduleM or an irreducibleprincipal series moduleM0 for the Levi-factor. However, if the left sidecorresponds toM, the right side corresponds toM. The only non-zero mapbetween these two has the trivial representation in the kernel and in thecokernel. Hence the only possibility is that the two sides are irreducibleprincipal series.

The condition b) is obtained from the fact that an irreducible principalseries forsl(2,R) always gives rise to certain local systemL over thesubmanifoldC∗ of P1. This local systemL does not extend toP1. Suchlocal systems which then arise in general and in connection with real roots,are described in [LV] and are always associated to a character defined asin b). ut(6.16) Proposition. AssumeG is a group with ample involutions. Letl(xsw) = l(x) + l(s) + l(w), s ∈ S, xw and xsw in WPPP. Assume thatO(xw) = O(xsw). Let g: J(ξ(w)) → J(ξ(sw))) and g(x) : J(ξ(xw)) →J(ξ(xsw)) the induced map. Then the induced map in homology

g(x) : HcodimO(xw)(n′, J(ξ(xw)))0→ HcodimO(xw)(n

′, J(ξ(xsw)))0

for n′ corresponding to a point inO(xw) is an isomorphism if and only if

a) g: Hq(n′′; J(ξ(w)))0→ Hq(n

′′; J(ξ(sw)))0 is an isomorphism for anyn′′corresponding to a point inO(w), equivalently;

b) w−1α is a real root, there is a mapΦ : SL(2,R)→ G whose image hasLie algebra generated by the root vectors corresponding tow−1α and−w−1α and for whichχ(ξ(w)(ΦDiag(−1,−1)) = −(−1)mw,αεα.

(6.17) Corollary. Let l(xsw) = l(x) + l(s) + l(w), s ∈ S, xw andxsw in WPPP. Assume thatO(xw) = O(xsw) is the openKKK orbit. Letg: J(ξ(w)) → J(ξ(sw))) and g(x) : J(ξ(xw)) → J(ξ(xsw)) the inducedmap. ThenJ(ξ(w)) → J(ξ(sw))) contains an asymptotically large repre-sentation in its image if and only if

a) g: Hq(n′′; J(ξ(w)))0→ Hq(n

′′; J(ξ(sw)))0 is an isomorphism for anyn′′corresponding to a point inO(w), equivalently;

b) w−1α is a real root, there is a mapΦ : SL(2,R)→ G whose image hasLie algebra generated by the root vectors corresponding tow−1α and−w−1α and for whichχ(ξ(w)(ΦDiag(−1,−1)) = −(−1)mw,αεα.

Proof. In the case ofG having ample involutions this is (6.16). WhenGdoes not have ample involutions then we apply (2.11). Note that sincethere are no arrowsa →→ b in these cases, necessarily the situationdescribed in b) does not take place. Thus (6.17) is satisfied vacuously in thesecases. ut


7. Main results

Axiomatic formulation ofC•Z.

In (5.9) we established the existence of a 2− act∗-acyclic cellular reso-lution, denotedC

•Z, overZ of KKK ×

KKK∩PPPXXX relative to{ZZZk}. We shall extract the

main features of this resolution and formulate them as axioms. LetB•

bea {ZZZk}-cellular resolution overR. We shall setQ(w) = KKK ×

KKK∩PPPSSS(w).

(7.1) Axioms.

a) act∗-acyclic.B•

is act∗-acyclic overR.

or

a1) 2− act∗-acyclic.B•

is 2− act∗-acyclic overR.

b) The sheaves are of Bruhat-type.

Bj =

⊕w∈WPPP

l(w)= j

B(w).

c) The maps are of BGG-type.There are mapsf(w,w′) : B(w)→ B(w′)compatible with squares, and such that withd(w,w′) = s(ξ(w), ξ(w′))f(w,w′) there is the commutative diagram

Bj −→ B

j ′x xB(w)

d(w,w′)−→ B(w′)

d) The sheaves are compatible with the Hecke algebra.If sa simple reflectionincreases the length ofw, thenB(w) = T(s)B(sw); likewise, if x increasesthe length ofy and sy, and x = sm . . . s1 is a reduced expression, thenB(sy) = T(s1) . . . T(sm)B(xsy). In addition, forx ∈ R(s)(Q(w)) the freepart of the stalks of the derived sheaves satisfies:Hq(B(sw))x,free = 0 ifq 6= codimQ(w) and x 6∈ Q(w); while HcodimQ(w)(B(sw))x,free = R ifx ∈ Q(w).e) The maps are compatible with the Hecke algebra.If there is a mapB(xy)→ B(xsy), andx increases the length ofy andsy, andx = sm . . . s1

is a reduced expression, then the mapB(y) → B(sy) is obtained byapplying the functorsT(s1) . . . T(sm) to the map fromB(xy) to B(xsy).


f) Reality.This condition is considered to be vacuously satisfied whenGdoes not have ample involutions (see (2.11) and (6.17)). Otherwise, weconsider the case thatG has ample involutions. Letw be inWP and lets bea simple reflection corresponding to a real root. Suppose that act∗B(w) andact∗B(sw) have the same support. LetO be aKKK -orbit dense in this supportand letx be in O. If the induced mapH0(act∗B(w))x→ H0(act∗B(sw))xis non-zero, then it is multiplication by±2. More generally, forw,w′ inWP, the induced mapH0(act∗B(w))x to H0(act∗B(w′))x is non-zero forx in theKKK -orbit O if and only ifw = xy,w′ = xsywith s andsyas above.If l(s1w) < l(w), l(s1w

′) < l(w′) then there is a local systemL on an openKKK orbit Q in T(s)O(w) with inclusion mapj : Q→ T(s1)O(w) such thatact∗B(w)⊗

ZQ = j!(L) = act∗B(w′)⊗

ZQ.

Remark.Each of the Axioms is motivated by certain results obtained inearlier sections. For example, in (7.1)d) the condition on the free part of thestalks comes from a sheaf version of (6.13). The version of these axiomswith a) we will call Strong Axioms; the version of these axioms with a1)instead of a) we refer to as Weak Axioms. These Weak axioms are to becomplemented with (9.9). In this paper we are able to produce a cellularresolution that satisfies the Weak Axioms. We fully expect that there isa stronger form of (4.16) from which one could conclude that the StrongAxioms are satisfied by this cellular resolution. The last statement in f)corresponds to 6.13 b) and c) as well as 2.12, 2.13, 2.14 and an easyanalogue of 2.14 which we did not state and which applies whenZ has thecharacter of an irreducible Verma module.

(7.2) Lemma. Let B•R be a cellular resolution overR of KKK ×

KKK∩PPPXXX rela-

tive to {ZZZk} and suppose that it satisfies the Axioms (7.1). Forp in theunique KKK -orbit dense in the support ofact∗B(w), the map induced onHcodimQ(w)(act∗B(w))p to HcodimQ(w)(act∗B(w′))p is non-zero if and onlyif it is multiplication by±2.

Proof.From (7.1)e) and base change, these maps are all obtained from mapsof the formH0(act∗B(y)[n])p→ H0(act∗B(sy)[n]p for somey, and withs in S increasing the length ofy. That is, writingw′ = xsywherew = xy,taking x = sk . . . s1 a reduced expression, applying the functorsT(x) =T(sk) . . . T(s1) to the derived sheaves foryandsy, and using base change oneobtains the mapHcodimQ(w)(act∗B(w))p to HcodimQ(w)(act∗B(w′))p. Theproof of the Lemma will be by induction onk. The casex = ebeing (7.1)f).

Fix p in the uniqueKKK-orbit dense in the support of act∗B(w). LetP1 = πsk

−1πsk(p) and take aKKK -orbit O that intersectsP1 on an open densesubset. Formπ−1

s πs(O) and let Z(xy) = O(xy) ∩ P1, and consider theinclusions denotedj(xy) : O(xy)→ π−1

s πs(O) and j(xy)P1 : Z(xy)→ P1.


LetLxy be the the zero cohomology sheaf of the restriction of act∗B(xy)[n+codimQ(w)] toO(xy). We haveLxy→ H0(Lxy)→ H0(Lxy)free= Lxy.The stalks ofLxy are free of rank one. Now the mapT(sk)act∗B(xy) →T(sk)act∗B(xsy), induces a mapM : Lskxy→ Lskxsy which, sincex′ = skxis of smaller length thanx, inductively is assumed to be±2.

We have an exact sequence of sheaves

T xy→ Lxy→ Lxy

whereT xy is a sheaf all of whose cohomology is torsion; similarly forxsy.We will consider below three pairs of chain complexes of sheaves ofZ

modulesA, B with a chain mapf : A→ B, namely:

act∗B(xy)≤ → act∗B(xy)

act∗B(xy)≤ → j(xy)!Lxy

j(xy)!Lxy→ j(xy)!Lxy.

The (. . . )≤ denotes a truncation functor so that chain complex on theleft and the chain complex on the right have the same sheaf cohomology indegrees less than or equal to 0. By the last part of 7.1 f) each of these mapsbecomes the identity map ifQ is used instead ofZ as coefficients. OverQ all these objects have the formj(xy)!Lxy ⊗

ZQ and thus they have their

cohomology in only one degree.Under these conditions each of the mapsf induces the identity map

(generator is sent to a generator) on the free part of all the cohomology

sheaves:Hq A freeHq( f)−→ HqBfree. These maps discussed are the vertical

arrows in the following commutative diagrams.

act∗B(xy) −→ act∗B(xsy)x xact∗B(xy)≤ −→ act∗B(xsy)≤

j(xy)!Lxy −→ j(xsy)!Lxsyx xact∗B(xy)≤ −→ act∗B(xsy)≤

j(xy)!Lxy×M1−→ j(xsy)!Lxswx x

j(xy)!Lxy −→ j(xsy)!Lxsy


We now applyT(sk) to all these diagrams and we have three commutativediagrams whose vertical arrows are all isomorphisms. These are obtainedby applyingH0(T(sk)(. . . )) free. We obtain from these that the map

H0(act∗B(skxy)) free−→ H0(act∗B(skxsy)) free

is equivalent to the map of sheaves of (free)Zmodules:

T(sk) j(xy)!Lxy×M1−→ T(sk) j(xsy)!Lxsw

Note, once more, that from the last part of 7.1 f) it follows that overQ,act∗B(xsy) and act∗B(xy) restricted to the copy ofP1 identified earlier,agree with j(xy)P1!Lxy ⊗

ZQ = j(xsy)P1!Lxsy⊗

ZQ. ConsequentlyT(sk)

applied to these complexes overQ is of the form j ′∗L′xy = j ′∗L

′xsy when

restricted to our copy ofP1. This implies that if the first map is non-zeroon stalks over the biggestKKK orbit in their common support, then the samething is true about second map.

Thus we obtain on aKKK-orbit (dense in the common support) that themapLx′y→ Lx′sy is multiplication by the integerM1 which occurs in themapLxy→ Lxsy. Hence from the inductive calculation,M1 = ±2. ut

(7.3) Theorem. C•Z, the geometric version of the BGG resolution, satisfies

the Weak Axioms (7.1).

Proof. The 2-acyclicity condition (a) is proved in (4.16) and (5.9). Thatconditions (b), (c), (d) and (e) are satisfied follows from the construction in(5.6) and the definition (5.7). For condition (f) notice that using base changeone gets the induced maps act∗CZ(w) → act∗CZ(sw) and act∗CZ(xy)→act∗CZ(xsy).

We suppose thatG is a group with ample involutions. Then most of (f)will follow from (6.16). In fact, it suffices to verify that ifw andsw areas in (7.1)d), then the corresponding complexes of sheaves act∗CZ(w) andact∗CZ(sw) contain the openKKK -orbit in their support, and that the mapsinvolved in (f) are given by multiplication by±2. Now forw andw′ as in(7.1)f), so thatw′ = sαw whereα is a real root (relative to a maximallysplit Cartan subgroup ofG), let XXXw andXXXw′ be the correspondingNNN-orbitsin GGG/BBB. Sinceα is a real root, there exists a Lie group homomorphismΦα : SL(2,R) → G. Then BBBwBBB ∪ BBBw′BBB containsΦαSL(2,R); henceXXXw ∪XXXw′ intersectsKKK/KKK ∩BBB. It follows that the support of act∗CZ(w) andof act∗CZ(w

′) contains the openKKK orbit in GGG/BBB.Fix x in the openKKK -orbit OOOo in GGG/BBB. Using smooth base change, the

cone of act∗CZ(w) → act∗CZ(w′) is R(s)act∗CZ(w). Denote byDw and

Dw′ the restrictions of the complexes of sheaves toP1 = π−1s πs(x). The

cone ofDw → Dw′ , C, is a complex of sheaves whose cohomology isconstant alongπ−1

s πs(x). Since on the stalks overOOOo, we have a map from


Z→ Z given by multiplication by an integern, the coneC is a complex ofsheaves non-zero only in degree one where it is a constant sheaf, with stalksZ/nZ. Using the notation of example (6.2) we have a distinguished triangle

T → Dw→ j∗L

whereT is a skyscraper sheaf all of whose cohomology is torsion. Considerthe long exact sequence in cohomology

· · ·→Hk−1(P1; j∗L)→Hk(P1;T )→Hk(P1;Dw)→Hk(P1; j∗L)→· · ·If k = 1 we obtain thatH1(P1;T ) = 0 andH1(P1;Dw) ∼= H1(P1; j∗L) ∼=Z/2Z. We also get thatHk(P1;T ) = 0 for k 6= 1. This puts us exactly inthe situation of example (6.2), son = 2.

If G does not have ample involutions then condition (f) is vacuouslysatisfied. utThe graph complexC•(G(B•R)).

We shall introduce a relation onWP that incorporates theR-form G.This will be a refinement of theBGGor Bruhat order.

(7.4) Definition. Let w and w′ be elements ofWP. We writew ⇒ w′provided:

a) w′ = xsy, w = xy, l(xsy) = l(x) + 1+ l(y), y ∈ WPPP, s ∈ S a simplereflection corresponding to the simple rootα.

b) y−1α is a real root and for the associated homomorphismΦα : SL(2, R)→ G, χ(ξ(y)(ΦDiag(−1,−1)) = −(−1)my,α εα.

Remark.Notice that ifw′ = xsαy andw = xy, thenw′ = sβw wherexsαx−1 = sβ. The apparent disagreement between the relation⇒ for thereal Schubert cells and the relation→ in theBGG-order, (1.1), is reconciledby the factξ reflects the length of an element, as is clear from the examplesin § 1.

(7.5) Definition.LetB•R be a complex of sheaves overRof KKK ×

KKK∩PPPXXX relative

to{ZZZk}which satisfies the Axioms (7.1). We define a graphG(B •R) consistingof a collection of vertices, one for each element ofWP. Two verticesw andw′ are connected with an oriented edge⇒ if and only if for x in the openKKK -orbit, the mapH0(act∗BR(w))x→ H0(act∗BR(w

′))x is non-zero.

We shall want as well to define a chain complex,C•(G(B•R)), associatedto the graphG(B•R). In order to define this we need to start with morefreedom in the choice of signs than given bys(ξ(x), ξ(y)) in the BGGresolution. A set of signse(·, ·) is a function defined onWP×WP such thate(ξ(x), ξ(y)) takes values in{0,±1} ande(ξ(x), ξ(y)) = 0 unlessx ⇒ y.


As with the BGG-signs, to obtain a chain complex one needs to imposea suitable compatibility. We note also that it is enough to definee(·, ·) onG(B•R) since on other edges thee is zero.

Given a set of signs{e(·, ·)} we attempt to form a chain complex,C•(G(B•R)), by associating to each vertexw of the graphG(B•R) a copy ofR, denotedRw. We define a map⊕

w∈WP

l(w)=k

Rw→⊕w′∈WP

l(w′)=k+1

Rw′

by the matrix(2e(ξ(x), ξ(y)). Our main assumption, at this stage, is thatein fact, gives rise to a chain complex. The existence of such signs will beproved in Lemma 7.8.

To simplify the notation, when the complex of sheaves isC•Z we shall

denote the oriented graph associated toC•Z simply byG. We caution the

reader that as we shall be computing cohomology it would be more naturalto use the notationG∗ as was used in the introduction for this graph.

Changing the BGG sign at one point:a) ConsiderWGGG as an oriented graph with edges→. Notice that if

γ is any path in this graph, consisting of a worda1 · · · an of (oriented)edges and inverses of edges, then aBGG sign function associates to ita number±1; namely the product of the signs along all the edges containedin the path. Denote this number ass(γ). This notation will be used 7.8.In particular if a = (w,w′) is any edge anda−1 its inverse, we will writes(a) = s(a−1) = s(w,w′).

b) We remark that anys(·, ·) can be changed as follows. Fix a vertexw.If w→ w′ or if w′ → w thens(w,w′) is multiplied by−1. The new signfunction satisfies (1.3) again. This simply corresponds to changing the signof one of the standard maps in theBGG resolution and then rearrangingsigns to compensate for this different choice so that the maps in the BGGresolution remain unchanged. We will generalize this procedure to any signfunctione(·, ·) in the proof of 7.8.

Making aBGGsign function positive along a tree.c) Given any treeT in WGGG, regarded as an oriented graph with respect

to→, the BGG sign s(·, ·) gives rise to anotherBGG sign sT(·, ·) whichis positive alongT. The modification ofs leading tosT is done by usingthe operation described in a) along the treeT. A vertexv in the treeT isfixed first. Then for any other vertexw in T, there is a unique path withinthe tree joiningv to w. This path can be represented by a word consistingof edges or inverses of edgesa1 · · · an. Here the inverse of an edge(x, y) is(y, x) and it can be interpreted as a path traveled in the opposite direction.Following the order given in this word, if we have already changed the signfunction alonga1 · · · ai into a new signs′ and onai+1 there is a negativesign assigned bys′, then the operation described in b) is performed relative


to the end point of the patha1 · · · ai+1. No incompatibilities may arise inthis procedure because no cycles are encountered.

d) Recall thatξ identifiesGwith a subgraph ofWGGG except thatξ reversesorientation:a⇒ b impliesξ(b)→ ξ(a). Thus given any subgraphZofGwewill obtain a subgraphξ(Z) of WGGG in which orientation has been reversed.Given maximal treesTi of each of the connected components ofG we mayapply the procedure of b) along each of theξ(Ti ). The result is aBGGsignfunctionsT, which is positive alongξ(Ti), whereT is the union of all theTi .We will say that such aBGGsign function has been normalized alongT (asubgraph ofG).

Notation: The cohomology with coefficients in a ringR attached to theoriented graph is the cohomology of this chain complex and is denotedH∗(G(B•R); R,e). In principle it depends on the signs chosen.

(7.6) Theorem. LetB•R be a cellular resolution overRof KKK ×

KKK∩PPPXXX relative

to {ZZZk} and suppose thatB•R andC

•R satisfy the Axioms (7.1). Then

a) if the Strong Axioms are satisfied, there is a choice of signse so that,

H∗(XC ; R) ∼= H∗(G(B•R); R,e).

b) If B•Z satisfies the Weak Axioms, then there is a surjection

H∗(G(B•R);Z,e)→ H∗(XC ;Z),and for anyc in the kernel,nc= 0 for some odd numberc.

Proof. We consider first case a) when the Strong Axioms of (7.1) are satis-fied. By 7.1 a)B

•R is act∗-acyclic. Then from Ia) of (4.13)H0(act∗B

•R[n])x

computesH∗(XC ; R), x in the openK -orbit, and agrees with the cellularchain complex{H p(Yp,Yp−1; R)}. In particular, referring to the definitionof act∗-acyclic in 4.12), which has a “torsion free” assumption, each term{H p(Yp,Yp−1; R)} of the cellular chain complex must also be torsion free.By assumption the complexC

•R also satisfies the Strong Axioms (7.1), from

the definition (4.12) and act∗-acyclicity in (7.1) it follows that for eachw inWPPP, H0(act∗C(w)[n])x ∼= R. Thus we obtain a chain complex of the form

· · · →⊕`(w)=k

Rw →⊕

`(w′)=k+1

Rw′ → · · · .

SinceB•R is of Bruhat-type we may choose a basisB by assigning to each

w an elementB(w) in Rw. From Lemma (7.2) we have that in this basis themaps betweenRw andRw′ will be zero or multiplication by±2. SinceB

•R

has maps ofBGG-type, (7.1)c), for an appropriate choice of signs e(·,·), thedifferential agrees with that inC•(G(B•R)).


If B•Z satisfies the Weak Axioms, then as in the statement and the proof

of 4.13II), there is a chain map corresponding to (c) of definition 4.12, con-sisting of a surjection term-by-term, from a chain complex with j-th termsH0(act∗(B

jZ[n]))x, free to a chain complex with cohomologyH∗(XC ;Z).

By 4.13 II), this chain map induces a surjection in cohomology with therequired property.

Therefore we obtain, once more, a chain complex of the form

· · · →⊕`(w)=k

Rw →⊕

`(w′)=k+1

Rw′ → · · ·

but now R = Z. The Weak Axioms are designed to ensure that this chaincomplex computesH∗(G;Z,e). For instance non-zero maps in the chaincomplex satisfy (7.4) because of (7.1) f) and are given by multiplicationby±2. ut

The determination of the choice of signseneeded for (7.6) will be solvedin 7.8.

(7.7) Corollary. For some family of signse there is a surjection

Hk(G;Z,e)→ Hk(G/P;Z)whose kernel consists of torsion classes killed by an odd integer.

Proof.We have from (7.3) that the geometricBGGcomplexC•Zsatisfies the

Weak Axioms; while from Proposition (4.6) and (4.16) we have that for somechoice of signse, there is the surjective mapHk(G;Z,e)→ Hk(G/P;Z)whose kernel consists of torsion classes killed by an odd integer. utRemark. From Bernstein’s Theorem (6.1) we know that the stalksH0(act∗CZ(w))x correspond ton-homology. ForG andP complex groupsand with coefficientsC, the Corollary gives Kostant’s well-known result onn-homology. Here too, the result is valid overZ for elementary reasons.Kostant’s more precise refinement in this case decomposing the direct suminto a sum over elements of lengthk in the Weyl group fails for real groupsandasymptotically largemodules, but (7.11) addresses this.

The intertwining operator complex.

The comparison of the cohomology ofG/P with that of the graphGmay be reformulated in other representation theoretic terms.

Let {πδ} be the set of asymptotically large principal series modulesof G with infinitesimal character that of the trivial representation, andin standard (submodule) position. We consider the representationsJ(w),defined in (1.11), parametrized byw ∈ WP. It is a standard fact thatHomU(g)(J(w),

⊕πδ) = HomU(g)(J(w), πδ) for someδ. Then for each

w ∈ WP we choose a generator of HomU(g)(J(w),⊕πδ) and denote it


R(w). The collection{R(w)} will be a basis for the complex. To definethe differential take the set of intertwining maps{ I BGG(w,w

′) : J(w) →J(w′)},w,w′ in WP,w→ w′, that are induced from aBGGfamily of mapsas in (1.4) or (1.5).

Definition. The collection of intertwining maps{R(w)} is said to becom-patibleif there is a set of signse(·, ·) such that

R(w′) I BGG(w,w′) = 2e(w,w′)R(w).

Starting with a collection of intertwining operators compatible withe(·, ·)and substitutingI BGG(w,w

′) = 12 I BGG(w,w

′) instead ofI BGG(w,w′)

we can eliminate the2’s and rewrite the compatibility conditions as follows:R(w′) I BGG(w,w

′) = e(w,w′)R(w).

Remark.Any set of signse arising from a compatible collection of in-tertwining operators automatically satisfies the requirements necessary todefine the cohomologyH∗(G;Z,e). A set of signse which arises froma compatible family of intertwining operators we will call anadmissiblefamily of signs.

Remark.i) TheBGGsigns give rise to aBGGfamily of maps as in section 1.The construction ofsT given previously can be carried out by changing thechoices (by a sign) of some of the standard maps that one starts with.

ii) Let F be a projective functor as in 3.8 of [BG] such thatF(M) isa Verma module wheneverM is a Verma module. Then application ofFgives aBGGfamily of maps associated to a different infinitesimal character.

iii) If F as in ii) is applied to the family of intertwining operators{R(w)}then a family of intertwining operators{FR(w)} is obtained. The compati-bility conditions become:FR(w′)F I BGG(w,w

′) = e(w,w′)FR(w).

iv) If w = ξ(σ) as in (1.11) thenL(F(U(g) ⊗U(p)

V[χ(ξ(σ)) ⊗ ζo])) =FJ(w). This is because projective functors commute withL. The new mapsF I BGG(w,w

′) are then obtained from theBGG family of maps which wasproduced by application ofF considering only maps associated to edgesin G. If F translates to infinitesimal character zero, then all the relevantmaps between generalized Verma modules are isomorphisms.

We define a chain complex4nt associated to a compatible collectionof intertwining maps,{R(w)}, with differentials defined by the family ofintertwining maps{ I BGG(w,w

′) : J(w)→ J(w′)}:

· · · →⊕σ∈WP

l(σ)=k+1

ZR(ξ(σ))∂k+1−−−−→

⊕τ∈WP

l(τ)=k

ZR(ξ(τ))→ · · ·


Dualizing term by term, we obtain the dual complex of intertwining oper-ators, roughly corresponding to 3.7 and denoted4nt; that is, we setZw =HomZ(ZR(w),Z) and we form the dual complex where4ntk = ⊕

l(w)=kZw.

As in iv) above, it is possible to “realize” the chain complex4nt in termsof intertwining maps between modulesFJ(w) and with maps obtained byapplication ofL to aBGG family of maps (along the edges ofG). The casewhenF translates to infinitesimal character zero will be used below.

(7.8) Lemma. (I) There exists a compatible family of intertwining operatorsand an admissible set of signse. There is also a corresponding surjection

Hk(G;Z,e)→ Hk(G/P;Z)whose kernel consists of torsion classesc such that for some odd integern,nc= 0.

(II) If the BGG sign s is normalized alongT, the union of maximaltrees in the various connected components ofG, (s = sT), then there isa compatible family of intertwining operators such that the correspondingsigne is a restriction toG of s extended by zero to other edges ofWP.

(III) Given any BGG family of maps and correspondingBGG sign s,there is a compatible collection of intertwining maps with admissible set ofsignse such thate= s on the edges ofG.

Proof. First we will prove (I), the existence of a compatible family ofintertwining operators. In order to do this we proceed to construct what wascalled above4nt in such a way that each4ntk is contained in a set of theform H0(n, . . . )0. By dualizing overZ we will have constructed a complexof the form 4nt with generators in each4ntk consisting of intertwiningoperators (contained in a set of the form HomC (H0(n, . . . )0,C) ). Recallthat the chain complex obtained from (4.13) part II with coefficients inZ isfree and is therefore contained, term by term, in the same construction with,say, rational coefficients.

From (7.3) we can conclude that usingZ coefficients means that a basishas been chosen in (4.13) part II overC with all the matrices having entriesin {0,±2}. We now use the representation theoretic interpretation of thechain complex in (4.13) a) overC. From Theorem (5.12) and (6.1) wecan conclude that the complex in (4.13) part II overC is in fact the samecomplex as the chain complex of (3.7). Thus the chain complex of (4.13) partII over Z, term by term, is obtained by choosing a basis in each summandof (3.7) so that the differentials are matrices whose entries are in{0,±2}.

The two functors Hom(. . . ;⊕πδ) and HomC (H0(n, . . . )0,C) agree onthe category of Harish–Chandra modules. UsingZ-coefficients in (4.13)part II (and dualizing) then provides us with a compatible family of inter-twining operators. This gives rise to a least one compatible family of inter-twining operators. For this compatible family the resulting chain complex


gives integral cohomology because of (4.13) b). The statement involvinga surjectionHk(G;Z,e)→ Hk(G/P;Z) follows from the same argumentof (7.7).

Now we prove (II). We fix a connected componentGi of G. Let T beany maximal tree in this graph. We can assume without loss of generalitythat along the treeT all the mapsI BGG(ξ(w), ξ(w

′)) involve positive signs.This amounts to normalizing the signe alongGi . We briefly describe thisprocedure which can then be extended to the whole graphG.

If e is not normalized as claimed, then we can change the intertwiningoperators{R(ξ(w))} (by a sign) inGi as follows. Letv denote a fixed vertexin Gi . If x = (w,w′) represents an edge in the tree, thenF(x) given bycomposition of an intertwining map with12 I BGG(ξ(x)) is a map sendinga generator to a generator, and is thus invertible on its domain which is thecorrespondingZ module of rank one. Lety = x−1 denote the edge(w′, w)viewed as a path traveled in the opposite direction. ThenF(y) denotes theinverse ofF(x). A path in the treeT determines a word involving edgesxin Gi or their inverses. Leta1 · · · am be such a word joining a vertexv toanother vertexw. Then by applyingF(a1) ◦ · · · ◦ F(am) to a generatorR(v)we obtain a fixed generator±R(w). Using these new intertwining operatorsour assumption that alongT all the signseare positive is satisfied. Ifx is anedge, recall that we can use the notations(ξ(x)) to denote the correspondingBGG sign; that is, ifx = (w,w′), s(ξ(x)) = s(ξ(w), ξ(w′)). Similarly weuse notatione(ξ(x)) for an admissible sign evaluated on an edgex. If γ isany path consisting of a word formed with edges and possibly inverses ofedges, we consider alsos(ξ(γ)) ande(ξ(γ)), the product of the signs of allthe edges involved.

Let b be any edge ofGi not in T. We will show inductively thate(ξ(b)) = s(ξ(e)). We proceed to obtain connected subgraphsTn con-tainingT, on whiche(x) = s(e) holds for all the edgesx. HereT1 = T.We now describe the inductive step. Assuming that onTn the two signfunctions agree, we constructTn+1. When b is added toTn, a cycleCcontained inTn ∪ { b} is produced. Leta1 · · · am be such a cycle andassumeaj = b. We first consider the case when this cycleC can beseen as consisting ofγ−1

1 γ2 , with eachγi a paths joining onew ver-tex to another vertexw′ of higher length and oriented so that it is al-ways going from vertices of smaller length to vertices of larger length.The edgeb is then contained in one of the two paths. These two paths,consisting of inclusions of generalized Verma modules, induce by com-position the same map except possibly for a sign. This sign is preciselythe products(a1) · · · s(am). Since mapsF(x) with x = (w,w′) are ob-tained functorially from the corresponding maps between two general-ized Verma modules, this translates easily intoF(a1) ◦ · · · ◦ F(an) send-ing a generator to plus or minus itself. Thus since only the edgeb isnot in Tn and on all edges inTn, s ◦ ξ, and e ◦ ξ agree, then neces-sarily e(ξ(a1)) · · · e(ξ(aj )) · · · e(ξ(am)) = s(ξ(a1)) · · · s(ξ(aj )) · · · s(ξ(am))and thereforee(ξ(aj )) = e(ξ(b)) = s(ξ(b)).


The general case presents an apparent difficulty because the cycleCmay involve inverses of edges (that isa = (w,w′) wherew′ → w). Notethat the condition on the two pathsγi that they are oriented towards in-creasing length, is only imposed because the maps in aBGG family ofmaps are not invertible. This lack of invertibility can be taken care of asfollows. We consider the projective functor from infinitesimal characterρ to infinitesimal character 0 as in [BG]. This functorF makes all theVerma modules involved in our cycle isomorphic. Thus the inverses of theproblematic edges oriented so that the length decreases, now correspond tomaps (isomorphisms) between generalized Verma modules. As in remarkiii) above this theorem, we obtain a collection of intertwining operatorsand FR(w′)F I BGG(w,w

′) = e(w,w′)FR(w). Now theF I BGG(w,w′) is

obtainable from aBGG family of maps as in ii) consisting only of iso-morphisms. This reduces us to the previous situation that was already dealtwith, but within a different infinitesimal character. We thus conclude thats= e.

For III) we consider an arbitraryBGG signs and constructsT in orderto apply I) and II). We obtain a compatible family of intertwining operatorswith an admissible sign that we denoteeT and which agrees withsT on theedges ofG. If we now reverse the process that producessT form s, thisamounts to changing the signs of some of the maps in theBGG family ofmaps associated tosT. Without changing the compatible collection of inter-twining maps, the admissible sign is changed because of the sign changesnow introduced in the mapsI BGG(w,w

′). What results is a compatible col-lection of intertwining maps such that the resulting admissible set of signsenow agrees withs on the edges ofG.

ut(7.9) Corollary. H∗(G/P;Z) and the cohomology of the dual complex ofintertwining operators have the same Betti numbers and the same summandsof the formZ/2sZ.

The complex of Schubert cells.

These isomorphisms of cohomology groups in fact may be realized atthe level of chain complexes. In § 4 a cellular resolution ofXC , denotedCell({Yp};Z), was obtained and, in Proposition (4.14), this was related tothe classical Bruhat cell complex of Schubert cells forGGG/PPP. Moreover,when restricted toX = G/P, it was shown in Proposition (4.15) to agreewith the Bruhat decomposition into real Schubert cells. This will providean isomorphism on the level of chains, relating the representation theoreticconstructions to the standard Schubert cells.

(7.10) Corollary. There is a chain map betweenG and the cellular chaincomplex associated to the real Schubert cells

Cell({Xp};Z) = · · · → H p(Xp, Xp−1;Z)→ H p+1(Xp+1, Xp;Z)→ · · · .


Here Xp denotes the union of allN+-orbits in G/P of dimension less thanor equal to p. This chain map induces a surjection in cohomology whosekernel consists of torsion classesc such thatnc= 0 for some odd integer.

Proof. Since C•Z satisfies the weak axioms (7.1) the complexH0(act∗C

•Z[n])x, freecomputes cohomology with a surjection toH∗(G/P;Z)

with the required properties. The cellular complex Cell({Yp};Z) free is de-fined in § 4 relative to the filtration{Yp = f −1{ZZZp ∩ act−1(x)}}, and isshown in Lemma (4.13) to agree with the complex of stalks, i.e.,G. Thischain map induces the isomorphism in cohomology.

From the inclusion ofK/K ∩ L ' [K ×K∩L

X] ∩ act−1(x) into its com-

plexification KKK/KKK ∩ LLL ' [[[KKK ×KKK∩LLL

XXX] ∩ act−1(x) we obtain a chain map,

namely restriction fromYp to Xp between the chain complexG and thecellular chain complex

· · · → H p((Xp, Xp−1;Z)→ H p+1(Xp+1, Xp;Z)→ · · ·ut

Homology and real Schubert cells.

For G and P complex groups, the graphG is totally disconnected, i.e.consists only of vertices indexed byWP and with no pair of vertices joinedby a⇒. Kostant’s result here that the cohomologyHk(G/P;C) is the sumover elements of lengthk in the Weyl group, becomes a sum over connectedcomponents ofG whose minimal length element has lengthk. Since co-homology classes and Schubert cells are in perfect duality here, in terms ofhomology the statement becomes that a Schubert cell is a cycle of dimen-sionk if and only the element inWP to which it is paired is minimal lengthin its connected component.

From this point of view, forG andP real groups we have

(7.11) Theorem. An N+ orbit c in G/P, represented byw ∈ WP, isa cycle if and only ifw has minimal length in its connected component inthe graphG∗.

Proof. As in (7.8) , the homology ofG/P is obtained from the chaincomplex associated to any compatible family of intertwining operators byapplying HomZ(. . . ,Z) to the complex. From the definition of a com-patible family, we know that the entries in the matrices representing thedifferentials are only non-zero when there is an arrow⇒ between twoelementsx, y in WP. More precisely there is a non-zero map betweentwo terms in the chain complexZR(ξ(x))→ ZR(ξ(y)) exactly when thereis an arrowy ⇒ x. This is a consequence ultimately of lemma (6.15)but it involves the dual of the map there. Thus we have a cycle corres-ponding tox ∈ WP exactly when there is no such arrow. This just means


that x must have minimal length along its connected component inG∗.ut

Remark.The statement in the Theorem is easily seen to equivalent tothe condition that there is noσ ∈ WP such thatσ ⇒ w. Or equiva-lently, a Schubert cellc is a cycle if and only ifc determines a non-zerointegral homology class[c]. This can be contrasted to Takeuchi’s resultthat overZ/2Z the Bruhat decomposition is a perfect cellular decompos-ition.

8. An example

(8.1) SL(4,R)/P, P a minimal parabolic subgroup.The vertices of thegraphG are given by Weyl elements in a minimal representation. A boxindicates that there is a trivial subquotient in that (generalized) princi-pal series representation. Several properties of the (co-)homology may beread-off from the graph: from the relation⇒, those (generalized) princi-pal series representations without the trivial representation as subquotientthat contribute to torsion in the cohomology; from the minimal elements inconnected components of the graph, the real Schubert cells that realize inte-gral classes; from the symmetry in the diagram, Poincare duality (includingtorsion).

Ex. SL(4,R)/Pmin

As elementary calculation from this graph produces the following resulton the cohomology groups.

i) H0(G/P,Z) = Zii) H1(G/P,Z) = 0iii) H2(G/P,Z) = Z/2Z⊕ Z/2Z⊕ Z/2Z


iv) H3(G/P,Z) = Z⊕ Z⊕ Z/2Z⊕ Z/2Zv) H4(G/P,Z) = Z/2Z⊕ Z/2Zvi) H5(G/P,Z) = Z/2Z⊕ Z/2Z⊕ Z/2Zvii) H6(G/P,Z) = Z

9. Frobenius eigenvalues and torsion

Here we consider the setting of [LV]. ThusGGG is a connected reductive groupoverFq, q = pr . We fixk = Fq and letθ denote an involutive automorphismof GGG. We assume thatKKK is a subgroup ofGGG having finite index in the fixedpoint set ofθ. We also consider a parabolic subgroupPPP as well as a BorelsubgroupBBB. The groupsBBB andKKK are all assumed to be defined overF(q).Assume thatp 6= 2 and fix a primel 6= 2. The notationGGG(k)means that weconsider the points ink of the corresponding algebraic group. Note that it ispossible to define in this setting, the counterparts of the groups with ampleinvolutions introduced in § 2. This is left to the reader.

We study theetale cohomology ofXk = KKK(k)/KKK(k) ∩ PPP(k). Instead ofSchubert cells inG/P we are then forced to consider the smooth varietiesccc(w) which are the intersections ofNNN(k)w−1BBB(k) with the openKKK(k) orbitin GGG(k)/PPP(k).We recall that overC these manifoldsccc(w) intersectG/P inN+ orbits as in (4.15). Moreover, intersection withG/P is such that cyclescorrespond to cycles (see (7.9) ). We will call these manifoldsccc(w) theSchubert manifolds overk. OverC these are complex manifolds that playthe role of theN+ orbits.

(9.1) Example.Let GGG = SLn,

GGG(k) = SLn(k) = { n× n matrices(xi, j )xi, j ∈ k,det(xi, j ) = 1}

and θ the involution defined byθ((xi, j ) = (xj,i )−1. Let PPP consist of the

upper triangular matrices inSLn(k). In this caseWP ∼= Sn the group ofpermutations inn letters. NowXk becomes the openKKK(k) orbit in the flagmanifold SLn(k)/PPP. For instance ifn = 2, Xk

∼= k \ {0} contained inP1 = k ∪ {∞}. The Schubert manifolds are sets consisting of one elementin k, andk with two points deleted respectively.

Using the results explained in [BBD] concerning reduction to positivecharacteristic, as well as a combination of (7.7) and (4.6), we obtain thefollowing theorem forQl etale cohomology. The proof of this theorem canalso be obtained directly overk by using the constructionL of section § 5which makes sense inetale cohomology over positive characteristic. Thisfunctor is applied to a geometric BGG resolution with coefficientsQl .


(9.2) Theorem. The cohomology ofXk with coefficients inQl is the co-homology of a chain complex

· · · →⊕w∈WP

l(w)=k

Ql,w→⊕w∈WP

l(w)=k+1

Ql,w→ · · · .

The differentials are matrices whose entries are integersa(x, y)with l(x) =k, l(y) = k+ 1 satisfying:

a(x, y) = 0 ⇔ x 6⇒ y

anda(x, y) = 2s(ξ(y), ξ(x)) If x⇒ y.

Frobenius eigenvalues attached to Schubert manifolds.

We now note that we can easily obtain information about eigenvalues ofFrobenius. Each copy ofQl,w in the above theorem comes with an eigenvalueof Frobenius given by a simple formula. Recall that the perverse sheavesthat correspond toJ(ξ(w) and J(ξ(sw) are related byDDDP (J(ξ(sw))) =T(s)DDDP (J(ξ(sw))). We also recall that the functorsT(s) give rise to theHecke algebra operatorTs and to an action of the Hecke algebra. Thisaction is described in [LV] very explicitly, and contains information on theFrobenius eigenvalues. Following the notation introduced in § 5 we denoteby D the set of all pairs of the formδ = (O,L) with O a KKK(k) orbit andLa KKK(k) equivariant local system onO. Let l(δ) = dimO. We also denote byδo the trivial sheaf on the openKKK(k) orbit in GGG(k)/BBB(k) andVδo the Verdierdual DDDδo of δo but now viewed as an element ofZ[q,q−1] ⊗

ZZ[D] as is

done in [LV]. This can be expressed asT−1x δ1 for certainx andδ1 a local

system on the openKKK(k) orbit extended by zero. If we apply the Heckealgebra operatorTw of [LV] to Vδo, then we obtain a combination of localsystems overKKK(k) orbits

TwVδo =∑

δ∈Dqz(δ)mδδ.

This equation, in the complex case withq = 1, gives an Euler characteristicof the perverse sheafDDDP (J(ξ(w))) in a Grothendieck group. Ifq = pr ,now in our current setting, then we have a corresponding Euler characteristicof a Ql perverse sheaf written in a Grothendieck group. The powers ofqindicate Frobenius eigenvalues attached to the local systems involved. Thereis oneδ(w) ∈ D in this expression such thatmδ(w) 6= 0 and ifmδ 6= 0 , thenl(δ) ≤ l(δ(w)). In fact in this casemδ(w) = ±1 . This implies that at mostone of the terms in the expression above consists of a local system over theopenKKK(k) orbit. The termqz(δ(w) is then the Frobenius eigenvalue attachedto the corresponding term in the chain complex in (9.2).


Various cases.

Recall that ifOOO is aKKK(k)-orbit, thenR(s)(OOO) is a union ofKKK(k)-orbits.We now assign notation to elements ofD that are supported inR(s)(O(w))and that occur in the expressionTwVδo.

a) SupposeR(s)(O(w)) consists of two orbits, both of which intersect thefibers ofπs in connected sets. Then there are two elementsδ1, δ2 in Dcorresponding to the two orbits involved. We are fixing the notation sothat dimδ2 < dimδ1 andδ2 is obtained by extending the local system toa KKK(k) equivariant local system onR(s)(O(w)) and then restricting tothe smaller orbit. There are two subcases. The first is whenδ(w) = δ1.The second subcase is whenδ(w) = δ2.

b) SupposeR(s)(O(w)) contains three orbits. We have two subcases. Thefirst subcase is whenδ(w) = γ andγ extends to aKKK(k) equivariantlocal systemγ on R(s)(O(w)). Let γ+ , γ− denote the correspondingrestrictions ofγ to the two smaller orbits. The second subcase is whenδ(w) = ε andε does not extend toR(s)(O(w)).

c) SupposeR(s)(O(w)) has two orbits, but one of them intersects the fibersof πs in two points. We haveδ(w) = η and a second element ofD, η1with dimη1 < dimη.

We now have a Hecke algebra analogue of (6.13) a) b) and c). Thisresults from a perverse sheaf analogue of (6.13) a) b) and c), which can beproved using (6.13) or directly working with perverse sheaves instead ofrepresentations. The perverse sheaf analogue of (6.13) says that ifl(sw) =l(w)+1 then on aP1 which is a fiber ofπs, pL(C(w)

Qlis of the form f∗(L)

where f is the inclusion ofO(w) into R(s)(O(w) andδ(w) = (O(w), L).These perverse sheaves restricted toR(s)(O(w)) are determined by theirrestriction to the fibers ofπs. We thus obtain perverse sheaves onP1. Theseperverse sheaves onP1 come with Frobenius actions and we obtain thefollowing calculation in the Hecke algebra module defined in [LV] :

(9.3) Lemma. Assuming thats is a simple reflection and thatl(sw) =l(w)+ 1, then the sum of the terms supported inR(s)(O(w)) which occurin the expressionTwVδo are of the form:

i) If δ(w) = δ1 as in case a), then we haveqz(w)[δ1+ (1− q)(δ2].ii) If δ(w) = δ2, as in case a), then we haveqz(w)[δ2].iii) If δ(w) = γ , as in case b), then we haveqz(w)[γ + (1− q)(γ+ + γ−)].iv) If δ(w) = η, as in case c), then we haveqz(w)[η+ (1− q)η1].v) If δ(w) = ε, as in case b), then we haveqz(w)ε.

Proof. We omit the proof since since it is similar to the proof of (6.13) butin the perverse category. ut


In the following lemma we use the notation introduced above,δ(w) andz(δ(w)).

(9.4) Lemma. Assuming thats is a simple reflection,w ∈ WP, l(sw) =l(w)+1, we havez(δ(sw)) = z(δ(w)) if and only ifw⇒ sw or dimO(w) <dim R(s)(O(w)). In the remaining casesz(δ(sw)) = z(δ(w))+ 1.

Proof. This follows by applying the operatorTs to each of the expressionsgiven above. The relevant formulas to applyTs are in lemma (3.5) of [LV].The case in (e) in that lemma occurs exactly when there is an arrow⇒. Thiscase corresponds to v) above. ut

Forw ∈ WP denote byL(w) the following variation of alength. Wewrite w = sm · · · s1 a reduced expression. Now there are exactlym1(w)of these simple reflections that arereal In this contextreal means thatdimT(si )O(si−1 · · · s1) = dimO(si−1 · · · s1). We setL(w) = m1 + [l(w)−m1]

2and letm2(w) be the number of arrows⇒ in the sequence:e→ s1 →s2s1→ · · · → sm . . . s1. we obtain inductively the following

(9.5) Proposition. We have for anyw ∈ WP z(δ(w)) = L(w) − m2(w).Therefore the numbersz(δ(w)) withw ∈ WP are constant along connectedcomponents of the graphG∗.

(9.6) Example.We have in example (9.1) that for anyn, L(w) = l(w) foranyw ∈ WP.

Recall thatz(δ(w)) is constant along the connected components in thegraphG. Each termw in the graphG has attached a Frobenius eigenvalue ofthe formqL(w)−m2(w). If we now dualize the chain complex of (9.2) to obtainhomology instead of cohomology then some of the elementsccc(w) whoseFrobenius eigenvalues we just described, combine to formcycles. In generalthe manifoldsccc(w) if they combine to form a cycle, give rise to a homologyclass inHl(w)(Xk; Ql), possibly zero, of Frobenius eigenvalueq−L(w)+m2(w).The action of Frobenius described is semisimple by the construction of thechain complex leading to that in (9.2). The construction can be carried outwith Z coefficients also and the chain complex of (9.2) is replaced witha chain complex involvingZl .

(9.7) Theorem.A Schubert manifold overk, corresponding to the parameterw ∈ WP is a cycle if and only ifw is of minimal length in a connectedcomponent of the graphG∗. The corresponding class inHl(w)(Xk; Ql) is aneigenspace of Frobenius with eigenvalueq−L(w)+m2(w).

Proof.This follows from the above discussion. ut(9.8) Example.We consider the case ofSL4(k) with θ as in (9.1), and referthe reader to the graphG∗ of example (8.1). We list all theZl cohomologywith (l, p) = 1; and this time, with an abuse of notation, in front of every


cohomology class we have written a Frobenius eigenvalueqL(w)−m2(w). OverQl the torsion disappears and theqz(δ(w)) become Frobenius eigenvaluesattached to cohomology classes attached to cohomology overQl .

i) H0(Xk,Zl) = q0Zl

ii) H1(Xk,Zl) = 0iii) H2(Xk,Zl) = qZl/2Zl ⊕ qZl/2Zl ⊕ qZl/2Zl

iv) H3(Xk,Zl) = q2Zl ⊕ q2Zl ⊕ q2Zl/2Zl ⊕ qZl/2Zl

v) H4(Xk,Zl) = q2Zl/2Zl ⊕ q3Zl/2Zl

vi) H5(Xk,Zl) = q3Zl/2Zl ⊕ q3Zl/2Zl ⊕ q3Zl/2Zl

vii) H6(G/P,Z) = q4Zl

(9.9) Proposition. For any summand of the formZ/psZ in the cohomologyof the chain complex

Cell({Yk},Z) free= · · · → H p(Yk,Yk−1;Z) free→ Hk+1(Yk+1,Yk;Z) free

→ · · ·necessarilyp= 2.

Proof. Using 4.16 and reduction to positive characteristic as described in[BBD], it is enough to prove that for anyl with (l,2) = 1 the chain complex

· · · → H p(Yk,Yk−1;Zl) free→ Hk+1(Yk+1,Yk;Zl) free→ · · ·cannot have anyp torsion if p is any odd prime. As in [LV] we considerXkwith p 6= 2. if p 6= 2 and(l, p) = 1 then there is a Frobenius action on thegroupsHk(Yk,Yk−1;Zl) and the quotientsHk(Yk,Yk−1;Zl) free. The actionon these groups is semisimple when the coefficients are changed toQl . EachWeyl group elementω in WP corresponds to an eigenspace of dimensionone where Frobenius acts with eigenvalueqL(ω)−m2(ω) with q a power ofthe primep. Therefore the action with coefficients inZl is also of the sameform. We conclude that there is an automorphism of the cohomology of

· · · → Hk(Yk,Yk−1;Zl) free→ Hk+1(Yk+1,Yk;Zl) free→ · · ·given by multiplication by a diagonal matrix with powers ofp in the diag-onal. This implies that there is nop torsion unlessp= 2. ut

With the aid of the previous result we can conclude our application ofrepresentation theory to the computation ofH∗(G/P;Z).(9.10) Theorem.There is a chain isomorphism between the chain complexassociated to a compatible family of intertwining operators

· · · →⊕w∈WP

l(w)=k

Zw→⊕w∈WP

l(w)=k+1

Zw→ · · ·


and the cellular chain complex

Cell({Yp},Z) free= · · · → H p(Yp,Yp−1;Z) free→ H p+1(Yp+1,Yp;Z) free

→ · · · .Restriction maps induce a quasi-isomorphismCell({Yp};Z)→ Cell({Xp};Z).

Proof.From (4.16) and (7.1) we know that the chain complex of intertwiningoperators and the cellular chain complex Cell({Yp},Z) free agree. Whilefrom (7.10) we know that the restriction map induces a surjection on thecohomology of Cell({Yp},Z) free to Cell({Xp},Z) with kernel consisting oftorsion classes killed by an odd prime. It follows from (9.9) that this map isan isomorphism. ut

Appendix to Sect. 2

Recall thatG(n) consist of all the permutationsx of (1, ...,n,−n, ...,−1)such that ifx(i) = − j then x(−i) = j . Each elementx of G(n) is de-termined by a permutationσ of (1, ..,n) and a sign,ε(x), so thatx(i) =ε(x)(i)σ(i).

If α is a root in∆+(G(n)) then a reflectionsα can be defined by firstintroducing a bilinear form< | >G(n) on the span oft1, .., tn by < λ1t1 +.. + λntn|λ′1t1 + .. + λ′ntn >= (λ1λ

′1 + ... + λnλ

′n). Then settingsα(λ) =

λ− 2<λ|α><α|α> α. Any sα then corresponds to one of the following:

a) exchange ofti andt j wherei < j , denotede(i, j) whenα = ti − t j

b) a sign changeti →−ti denoteds(i) whenα = tic) x = s(i)s( j)e(i, j) whenα = ti + t j .

Fix anyk and denotes(i, k) = 1 if i < k ands(i, k) = −1 if i > k. Wewill also denote by|Z| the number of elements of any finite setZ.

(A.1) Lemma. Let y = sα and consider the casesα = e(i, j). Then∆R3 = 0.

Proof.An exchange ofti andt j does not change the signs of any of the rootsof the form±tk and thus, by definition∆R3 = 0. ut(A.2) Lemma. Let y = sα and consider the casesα = s(i). Then(∆R1 +∆R2)∆R3 ≥ 0.

Proof.Fix anykand consider the set of two positive rootsA={s(i, k)(ti − tk),ti + tk}. If we apply an elementw ∈ G(n) to this set, corresponding toa permutationσ and a sign,ε(w), we obtainB = {s(i, k)(ε(w)(i)tσ(i) −ε(w)(k)tσ(k)), ε(w)(i)tσ(i)+ε(w)(k)tσ(k))}. We now apply the sign changes(i)


and obtain a third setC = { −s(i, k)(ε(w)(i)tσ(i) − ε(w)(k)tσ(k)),−ε(w)(i)tσ(i) + ε(w)(k)tσ(k))}. Now the contribution to∆R1 + ∆R2 from this indi-vidual case is|C| − |B|. To prove our lemma it is then enough to compute|C| − |B| in all the possible cases that will arise. We have the followingcases to considerk < i or i < k and, for each of these, all the possiblesigns ε(w)(i) = ±1, ε(w)(k) = ±1 and then we will still have eitherσ(k) < σ(i) or σ(k) > σ(i). One can now directly check that in all these16 cases(∆R1 + ∆R2)∆R3 ≥ 0. We leave this easy case by case analy-sis to the reader. Note that∆R3 = 1 whenε(w)(i) = 1 and∆R3 = −1whenε(w)(i) = −1. Then it turns out, by inspection of all the 16 cases, that|C|−|B|has absolute value 0 or 2. Moreover|C|−|B| is non-zero only in thecases when (i)i < k and (ii)σ(i) < σ(k). Furthermore,|C| − |B| = 2 whenin addition to the previous conditions (i) and (ii) we also haveε(w)(i) = 1and ε(w)(k) = −1 (hence∆R3 = −1). Also |C| − |B| = −2 when inaddition to (i) and (ii) we haveε(w)(i) = −1 (hence∆R3 = 1). ut

(A.3) Lemma. Let y= sα and consider the casesα = s(i)s( j)e(i, j). Then(∆R1 +∆R2)∆R3 ≥ 0.

Proof. First fix anyk different from eitheri or j and consider first the twoset of two positive rootsA(m) = {s(m, k)(tm− tk), tm+ tk}, with m= i, j .We apply anyw determined by a permutationσ and a signε(w) andthus obtainB(m) = {s(m, k)(ε(w)(m)tσ(m)− ε(w)(k)tσ(k)), ε(w)(m)tσ(m)+ε(w)(k)tσ(k))} with m = i, j . Next we applys(i)s( j)e(i, j) and obtainC(m) = {s(m, k)(−ε(w)(m)tσ(w)(m′) − ε(w)(k)tσ(k)),−ε(w)(m)tσ(m′)+ ε(w)(k)tσ(k))} with m = i, j and, wherem′ = i if m = j , andm′ = j ifm= i . Now eachk contributes to∆R1+∆R2 by |C(i)|−|B( j)| +|C( j)|−|B(i)|. There is a list of cases that must be considered. First there are threepossibilities:k < i < j , i < k < j , i < j < k. In each of these threeall possible combinations of signs occur:ε(w)(k) = ±1, ε(w)(i) = ±1,ε(w)( j) = ±1. However, as we will observe below, it is enough to considerthe cases whenε(w)(i) andε(w)( j) agree, that is, only four different pos-sibilities. For each combination of signs being considered there are still thefollowing cases:σ(m) < σ(k) for m = i, j or σ(m) > σ(k) for m= i, j orσ(i) < σ(k) andσ( j) > σ(k) or, finally, σ(i) > σ(k), σ( j) < σ(k). This isa total of 48 cases.

We observe that∆R3 is positive exactly whenε(w)(i), ε(w)( j)are equalto 1. In this case∆R3 = 2. Also ∆R3 is negative whenε(w)(i), ε(w)( j)are both equal to−1 and in that case∆R3 = −2. Whenε(w)(i), ε(w)( j)have different signs then∆R3 = 0 and there is nothing to prove. Becauseof these all these cases whenε(w)(i) 6= ε(w)( j) can be ignored.

We leave to the reader the easy but lengthy verification that in each of the48 cases, (where∆R3 6= 0), one has|C(i)| − |B( j)| + |C( j)| − |B(i)| > 0only whenε(i) = ε( j) > 0 and|C(i)| − |B( j)| + |C( j)| − |B(i)| < 0 onlywhenε(i) = ε( j) < 0.


In addition we still have the family of cases whenk = j . We considerA = {ti − r j ), ti + r j } and we applyw. There are then the casesε(w)(i) =±1 andε(w)( j) = ±1, but, as before, it is enough to examine the casesε(w)( j) = ε(w)(i) and there are only two possibilities. For each of these,we still have the casesσ(i) < σ( j) andσ(i) > σ( j). We thus have a total of4 cases to consider. From here, as before, we obtain two setsB andC andthe contribution to∆R1+∆R2 is |C|− |B|. Here, as before, a case-by-caseanalysis shows that if∆R3 6= 0, which meansε(w)(i) = ε(w)( j), then|C| − |B| > 0 impliesε(i) = ε( j) = 1 and the condition|C| − |B| < 0impliesε(i) = ε( j) = −1. ut

We obtain as a consequence of these three lemmas, with notation as inA.1, A.2, A.3

(A.4) Proposition. Let y = sα whereα ∈ ∆+G(n). Then(∆R1 + ∆R2)∆R3 ≥ 0.

Proof.This follows from A.1, A.2 and A.3. ut(A.5) Lemma. Let < | > be a G(n) invariant non-degenerate bilinearform on theR vector space spanned byti i = 1, ..,n by the actionx(ti ) =ε(x)(i)tσ(i) for anyx ∈ G(n). Then< | >= N< | >G(n) for some non-zeroscalar N.

Proof.Using the definition of< | >G(n) It suffices to show that< ti |t j >= 0for i 6= j and that all the scalars< ti |ti > agree. Since the bilinear formis non-degenerate all these numbers< ti |ti > will be non-zero. We firstapply a sign changes(i) to < ti |t j > with j 6= i . We have byG(n)invariance:< −ti | t j >=< ti | t j >. Thus< ti |t j >= 0 for i 6= j . Usingan exchangee(i, j) we also have that< e(i, j)ti ,e(i, j)ti >=< ti |ti > thus< t j |t j >=< ti |ti > for anyi, j . ut

Let λ1t1 + ... + λntn = λ. Recall that ifG = G(a,b, c) as in 2.11then we implicitly assume that there is a positive system ofh roots suchthat the corresponding restricted roots agree with∆+(G(n)) except formultiplicities. For anyβ a positive root ofG with respect toh denote byα ∈ ∆+(G(n)) its restricted root, and letTβ(λ) = λ − (rβ) < λ|α > αwhererβ > 0 and< | > is any G(n) invariant non-degenerate bilinearform.

(A.6) Lemma. LetG = G(a,b, c) and letβ be a positive root with respectto a positiveh system compatible with∆+(G(n)). If there isx ∈ G(n) suchthat for all λ we haveTβλ = x(λ), x 6= e (the identity), then for someα ∈ ∆+G(n) we havesα = Tβ. In particular Tβ is of the forms(i) or of theform e(i, j) or of the forms(i)s( j)e(i, j).

Proof.We have three cases depending on the restricted roots correspondingto β. First assume that the restricted rootα associated toβ is ti − t j . If we


applyTβ to an arbitraryλ = λ1t1 + ...+ λntn then the coefficientλi of ti issent to a scalar of the form(1−a)λi+aλ j for somea 6= 0 (a= Nrβ, N as inA.5), andλ j is sent to(1−a)λ j +aλi . This already uses A.5 which gives usthe general shape of aG(n) invariant non-degenerate bilinear form. All theother coefficients are unchanged. SinceTβ is not the identity, and for anyλthe scalar(1− a)λi + a λ j must be of the form±λk, the only possibilityis thata = 1. Thus we obtainTβ = e(i, j). The other cases are similar sowe just sketch them. In the case whenα has the formti + t j , Tβ sendsλito (1− a)λi − a λ j andλ j to (1− a)λi − a λi . Once more it follows thata = 1 and thusTβ acts likes(i)s( j)e(i, j). Then there are the cases whenα has the formδti whereδ = 1,2. In this casesλi is sent to(1− a)λi forsome positivea and necessarilya= 2 giving rise to a sign changes(i). ut

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Schubert cells and representation theory