Quotient of Deligne-Lusztig varieties

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Olivier Dudas^∗^† December 20, 2011

Abstract

We study the quotient of parabolic Deligne-Lusztig varieties by a finite unipotent groupU^F whereUis the unipotent radical of a rational parabolic subgroup P=LU. We show that in some particular cases the cohomology of this quotient can be expressed in terms of "smaller" parabolic Deligne- Lusztig varieties associated to the Levi subgroupL.

Introduction

The very first approach to the representation theory of finite reductive groups is the construction of representations via Harish-Chandra (or parabolic) induction. IfGis a connected reductive group over F=F_p with anF_q-structure associated to a Frobenius endomorphism F:G−→G, andPis anF-stable parabolic subgroup with anF-stable Levi complementL, one can define, over any ringΛ^, the following functors

and

R^G_L :Λ^L^F^-mod^−→Λ^G^F^-mod

∗R^G_L :Λ^G^F^-mod^−→Λ^L^F^-mod

called Harish-Chandra induction and restriction functors. One of the main fea- ture of these functors is that they satisfy the so-called Mackey formula: if Q is another F-stable parabolic subgroup withF-stable Levi complementMthen

∗R^G_M◦R^G_L ≃ X

R^L_L∩xM◦^∗R^x_L∩^MxM◦adx

where x runs over a explicit finite set associated to L and M. In addition to being a powerful tool for studying an induced representation, this formula is also essential for proving that the Harish-Chandra functors depend on L only and not on the choice of P.

It turns out that not all the representations ofG^F can be obtained by Harish- Chandra induction (already forG=SL₂(F), many representations arecuspidal).

∗Oxford Mathematical Institute.

†The author is supported by the EPSRC, Project No EP/H026568/1 and by Magdalen College, Oxford.

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To resolve this problem Deligne and Lusztig defined in [6] a generalised induction in the case where P is no longer F-stable but L still is. They constructed morphisms between the Grothendieck groups

and

R^G_L :K₀(Λ^L^F^-mod)^−→^K0(Λ^G^F^-mod)

∗R^G_L :K₀(Λ^G^F^-mod)^−→^K0(Λ^L^F^-mod)

still satisfying the Mackey formula. These morphisms come from a virtual char- acter given by the ℓ-adic cohomology of a quasi-projective variety Xe_L,P, the parabolic Deligne-Lusztig variety associated to (L,P). Here, Λ is a finite extension ofQ_ℓ,Z_ℓorF_ℓ.

WhenΛis a finite extension ofQ_ℓthe categoryΛ^G^F-mod is semisimple, and its Grothendieck group encodes most of the information. However, in the modu- lar framework, that is whenΛ⁼^ZℓorF_ℓ, the Deligne-Lusztig induction and restriction morphisms give only partial information on the category of modules. To obtain homological properties, one needs to consider the complex RΓ^c^(X,Λ^{) rep-} resenting the cohomology of the variety in the derived category D^b(Λ^G^F^-mod).

Using this point of view, Bonnafé and Rouquier defined in [1] triangulated functors

and

R^G

L⊂P :D^b(Λ^L^F^-mod)^−→^D^b⁽Λ^G^F^-mod)

∗R^G

L⊂P :D^b(Λ^G^F^-mod)^−→^D^b⁽Λ^L^F^-mod).

Unlike the previous functors, these are not expected to satisfy a naive Mackey formula as they highly depend on the choice ofP. However, there is a good evi- dence that the composition^∗R^G

M⊂Q◦R^G

L⊂Pshould be somehow related to functors associated to smaller Levi subgroups. The purpose of this paper is to investi- gate the case where Qis F-stable. IfU denotes its unipotent radical, then the composition ^∗R^G

M⊂Q◦R^G

L⊂P is induced by the cohomology of the quotient variety U^F\eXL,P.

In the original paper of Deligne and Lusztig [6], the Levi subgroupL is a torus and XeL,P corresponds to some element w of the Weyl groupW of G. The motivating example is when (L,P) represents a Coxeter torus, that is whenwis a Coxeter element ofW. In that case, the variety X_L,P=Xe_L,P/L^F is contained in the maximal Schubert cell and its quotient byU^F has been computed by Lusztig in [13]. In the case whereΛ⁼^Qℓit is given by the following quasi-isomorphism ofM^F-modules:

RΓc(U^F\X_L,P,Q_ℓ)≃RΓc(X_L∩M,P∩M,Q_ℓ)⊗RΓc((F^×)^d,Q_ℓ)

where d is the semisimple Fq-index of M in G. Surprisingly, this isomorphism does not come from a M^F-equivariant isomorphism of varieties, and we will see that it is more natural to study the quotient ofXe_L,P instead of X_L,P.

In general, the variety XL,P is not contained in only one Schubert cell. The strategy towards the determination of the cohomology ofU^F\Xe_L,Pwill consist in the following steps:

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• decompose the varietyXeL,P intopiecesXex coming from the decomposition ofG/PintoQ-orbits (see Section2);

• in some well-identified cases, express the cohomology ofU^F\Xxin terms of parabolic Deligne-Lusztig varieties associated to Levi subgroups ofM(see Section3).

The second step is undoubtedly the most difficult. We are able to provide a sat- isfactory solution to this problem in presumably very specific situations, namely when the pair (L∩^xM,P∩^xM) is close to (L,P) (see Theorem3.11for more details). However, it turns out that our main result is general enough to cover most of the Deligne-Lusztig varieties associated to unipotent Φd-blocks with cyclic defect group. This should give many new results on the geometric version of Broué’s abelian defect conjecture. To illustrate this phenomenon, we compute in Section 3.3 the principal part of the cohomology of the parabolic variety associated to the principalΦ2n−2-block for a group of typeB_nas well as its Alvis-Curtis dual. In subsequence papers this baby example will be supplemented by the following more involved results:

• for exceptional groups, the determination of the cohomology of varieties associated to principal Φd-blocks when d is the largest regular number besides the Coxeter number. This should be enriched with predictions for the corresponding Brauer trees;

• for groups of type A_n, the determination of the cohomology of varieties associated to any unipotent block from the knowledge of the cohomology of the variety X(w²₀).

1 Parabolic Deligne-Lusztig varieties

LetGbe a connected reductive algebraic group, together with an isogeny F, some power of which is a Frobenius endomorphism. In other words, there exists a positive integerδsuch thatF^δ defines a splitF_qδ-structure onGfor a certain power q^δ of the characteristic p (note that q might not be an integer). For all F-stable algebraic subgroupHofG, we will denote byHthe finite group of fixed pointsH^F.

We fix a Borel subgroup B containing a maximal torus T of G such that both Band Tare F-stable. They define a root sytem Φ ^{with basis}∆, and a set of positive (resp. negative) roots Φ⁺ ^(resp. Φ⁻). Note that the corresponding Weyl groupW is endowed with an action ofF, compatible with the isomorphism W ≃N_G(T)/T. The set of simple reflections will be denoted by S. We shall also consider representatives { ˙w|w∈W}ofW in N_G(T) compatible with the action of F (this is possible by [9, Proposition 8.21]).

To any subsetI⊂Sone can associate a standard parabolic subgroupPI containing B and a standard Levi subgroup LI containing T. If UI denotes the unipotent radical of PI, the parabolic subgroup can be written as PI =LIUI.

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Let U (resp. U⁻) be the unipotent radical of B (resp. the opposite Borel subgroup B⁻). Each root α defines a one-parameter subgroup U_α, and we will denote by u_α :F−→U_α an isomorphism of algebraic group. In order to sim- plify the calculations, we shall choose these isomorphisms so that u_α(λ) ˙s_α = u_−α(λ⁻¹)α^∨(λ)u_α(−λ⁻¹). Note that the groupsU_α might not be F-stable in gen- eral even though the groupsUandU⁻ are.

Finally, we denote by B⁺_W (respB_W) the Artin-Tits monoid (resp. Artin-Tits group) of W, and byS={sα|α∈∆^} its generating set. The reduced elements of B⁺_W form a setWwhich is in bijection withW via the canonical projectionBW։ W. We shall also consider the semi-direct productB_W⋊_〈F〉_where _F_·_b₌^F_b_·_F_.

Let Ibe a subset of Sand denote by B⁺_I the submonoid ofB⁺_W generated by I. Following [9], we will denote by I−→^b ^FI any pair (I,b) with b∈B⁺_W satisfying the following properties:

• any left divisor ofbinB⁺_I is trivial;

• ^bFI=I, that is everys∈Isatisfiesb⁻¹sb∈^FI.

Digne and Michel have constructed in [9] a parabolic Deligne-Lusztig variety X(I,bF) associated to any such pair. Note that when b=w∈W and if w denotes its image by the canonical projectionB_W։^W, the previous conditions are equivalent to wbeing I-reduced and^wFI=I. In that case, the variety X(I,wF) can be written

X(I,wF)=©

g∈G¯¯g⁻¹^Fg∈PIw^FPIª±

PI.

As in the case of tori, we can construct a Galois covering of X(I,wF). It is well- defined up to a choice of a representative nofwin N_G(T):

X(e I,nF)=©g∈G¯¯g⁻¹^Fg∈UIn^FUIª±

UI.

The natural projectionG/UI−→G/PI makesX(e I,nF) aL^nF_I -torsor over X(I,wF).

By using an F-stable Tits homomorphismt:B_W−→N_G(T) extendingw∈W7−→

w, Digne and Michel have generalised in [9] this construction to any element˙ I−→^b ^FI. The corresponding variety will be denoted byX(I,e bF). It is a L^t(b)F_I - torsor over X(I,bF). Whenb=w∈Wwe shall simply denotet(w) by ˙w.

Remark 1.1. When I is empty, we obtain the usual Deligne-Lusztig varieties X(bF) andX(bFe ) associated to any elementbof the Braid monoid (as defined in [3] or [1]).

2 Decomposing the quotient of X(I,wF)

Let (I,w) be a pair consisting of an elementwofW and a subset IofS such thatwis I-reduced and^wFI=I. Let J be another subset ofS. If J isF-stable, then so is the corresponding standard parabolic subgroup PJ and its unipotent radical UJ. In this section we are interested in describing the quotient of the parabolic Deligne-Lusztig variety

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X(I,wF)=©g∈G|g⁻¹^Fg∈PIw^FPIª±

PI

by the finite unipotent groupUJ. Our main goal is to express this quotient (or at least its cohomology) in terms of "smaller parabolic varieties" associated to the Levi subgroupLJ.

Throughout this paper, Λwill be any extension of the ringZ_ℓofℓ-adic inte- gers. We shall always assume thatℓ is different from p, so that by cohomology over Λwe mean the extension of the étale cohomology of quasi-projective varieties with coefficients in Z_ℓ. The properties of RΓc(−,Λ) that we will use are either classical or can be found in [14].

2.1 A general method

Recall that the partial flag variety G/PI admits a decomposition into PJ- orbitsG/P_I=`P_JxP_Iwherexruns over any set of representatives ofW_J\W/W_I. The restriction of this decomposition to X(I,wF) can be written as

X(I,wF)= a

x∈[WJ\W/WI]

©pxP_I∈PJxP_I/PI¯¯p⁻¹^Fp∈x(P_Iw^FPI)^Fx⁻¹ª

. (2.1) We will denote by Xx=X(I,wF)∩PJxP_I a piece of this decomposition. It is a locally closed P_J-subvariety of X(I,wF). Now, each of these pieces can be lifted up toPJ. More precisely, if we define the variety

Zx =©p∈PJ¯¯p⁻¹^Fp∈x(P_Iw^FPI)^Fx⁻¹ª

then the canonical projection G−→G/P_I induces a fibration Z_x−→X_x with fiber isomorphic toPJ∩^xPI. Now if we assume thatxisJ-reduced-I, the intersection PJ∩^xPI can be decomposed as PJ∩^xPI =(LJ∩^xPI)·(UJ∩^xU). Furthermore, L_J∩^xP_I is a standard parabolic subgroup ofL_J (it containsL_J∩B) and hence it can be writtenLJ∩PK_x withKx=J∩^xΦI. The cohomology of Xx is thus given by

RΓc(X_x,Λ⁾^≃^RΓc(Z_x/L_J∩P_K_x,Λ^{)[2 dimU}J∩^xU]. (2.2) The advantage of this description is that the quotient of the variety Zx byU_J is easier to compute. If we decompose p∈PJ as p=ul∈UJLJ then the quotient variety can be written (see for example [11, Proposition 1.3])

U_J\Zx=©

( ¯p,l)∈£(xP_Iw^FPIFx⁻¹)∩PJ¤

×LJ¯¯πJ( ¯p)=l⁻¹^Flª whereπJ:PJ−→LJ is the canonical projection.

Our aim is to relate this variety to "smaller" parabolic Deligne-Lusztig varieties. For that purpose, we need to identify the double cosets in whichl^−1Fllies, which amounts to decomposing the intersection (xP_Iw^FPIFx⁻¹)∩PJ as well as its image underπJ. Letv∈W_Jbe aK_x-reduced-^FK_xelement. We can decompose the double coset P_K_xv^FP_K_x as follows:

PK_xv^FPK_x=(LJ∩PK_x)UJv(L_J∩^FPK_x).

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SinceLJ∩PKx=LJ∩^xPIis a subgroup of^xPI, the intersection (xP_Iw^FPIFx⁻¹)∩ (PK_xv^FPK_x) is non-empty if and only if (xPIw^FPIFx⁻¹v⁻¹)∩UJ is. In this case, the projectionπJ:PJ−→LJinduces a fibration (xPIw^FPIFx⁻¹)∩(PK_xv^FPK_x)−→

(LJ∩PKx)v(L_J∩^FPKx) with fiber isomorphic to (xP_Iw^FPIFx⁻¹v⁻¹)∩UJ. If we define Z^v_x to be the variety

Z^v_x =©

( ¯p,l)∈£(xP_Iw^FPIFx⁻¹)∩(PK_xv^FPK_x)¤

×LJ¯¯πJ( ¯p)=l⁻¹^Flª

then we obtain a decomposition ofUJ\Zxinto locally closed subvarieties together withL_J-equivariant maps

Z^v_x−→©l∈LJ|l⁻¹^Fl∈LJ∩PKxv^F(LJ∩PKx)ª

(2.3) with fibers isomorphic to (xP_Iw^FP_I^Fx⁻¹v⁻¹)∩U_J.

Remark 2.4. In the case where^vFK_x=K_x, the quotient by L_J∩P_K_x of the variety on the right-hand side of 2.3can be identified with the parabolic Deligne- Lusztig variety associated toK_x−→^v ^FK_x. We shall, by convenient abuse of nota- tion, denote it by X_L_J(K_x,vF) even whenvF does not normaliseK_x.

Finally, we setZ^v_x=Z^v_x/L_J∩P_K_x. The right action ofU_J∩^xUon Z_xinduces an action byF-conjugation onZ^v_x and letX^v_x=Z^v_x/UJ∩^xUI be the quotient (equiva- lently, it is the image of Z^v_xby the morphismU_J\Zx։^U^J^\X^x). At this point we have obtained

• A decomposition ofU_J\X(I,wF) into some locally closedL_J-varietiesX^v_x.

• A quasi-isomorphism RΓ^c⁽X^v_x,Λ⁾^≃^RΓ^c⁽Z^v_x,Λ^{)[2 dimU}^J^∩^xU] (obtained as in2.2).

• A L_I-equivariant morphism Z^v_x −→XL_J(K_x,vF) with fiber isomorphic to (xPIw^FPIFx⁻¹v⁻¹)∩UJ.

Therefore, if we want to express the cohomology ofU_J\X(I,wF) in terms of the different varieties X_L_J(K_x,vF) that can appear we need to refine the description of the latter morphism. This will be done in Section2.3after discussing the case of parabolic varieties associated to elements of the Braid monoid.

Remark 2.5. When ^vFK_x=K_x, we can actually be more precise: l⁻¹^Fl can be written uniquely asl₁v˙^Fl₂withl₁∈(LJ∩UK_x)∩^vF(LJ∩U⁻_K

x) andl₂∈LJ∩PK_x. Then for z∈(xP_Iw^FPIFx⁻¹v⁻¹)∩UJ we have (l₁zv˙^Fl₂,l)∈Z^v_x and all the elements are obtained that way. In other words, we have the following isomorphism of varieties

Z^v_x ≃£(xPIw^FPIFx⁻¹v⁻¹)∩UJ¤

×©l∈LJ|l⁻¹^Fl∈LJ∩PK_xv^F(LJ∩PK_x)ª . Through this isomorphism the group L_J (resp. LJ∩PK_x) acts on l∈LJ by left (resp. right) multiplication. However, it is more difficult to describe the action of LJ∩PK_x on (xPIw^FPIFx⁻¹v⁻¹)∩UJ. In particular, Z^v_x is in general not isomorphic to £(xP_Iw^FPIFx⁻¹v⁻¹)∩UJ¤

×X_L_J(K_x,vF). We shall nevertheless give many examples where the cohomology of these two varieties coincide.

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2.2 Elements of the Braid monoid

By [9, Section 6] any elementI−→^b ^FIcan be decomposed asI=I₁−→^w¹ I₂−→ · · ·^w²

wr

−→I_r+1 =^FI where w_i ∈W. Using this property one can easily generalize the previous constructions to X(I,bF): to each tuple x= (x₁, . . .,xr) with xi a J- reduced-I_i element ofW one can associate varieties X_xand Z_xsuch that

Zx= (

(p₁, . . .,p_r)∈(PJ)^r

¯¯

¯

p⁻¹_i p_i+1∈x_iP_I_iw_iP_I_i+1x⁻¹_i+1 p⁻¹_r ^Fp₁∈xrPI_rwrFPI₁Fx⁻¹₁

)

and RΓc(Xx,Λ⁾^≃^RΓc¡ Zx±Q

LJ∩PK_xi,Λ^¢£²^P^dim^UJ∩^xⁱU¤ withK_x_i=J∩^xⁱΦIi.

By looking at the intersections of x_iP_I_iw_iP_I_i+1x⁻¹_i+1 with double cosets of the form PK_xiviPK_xi+1 one can decomposeUJ\Z_x into locally closed subvarieties Z^v_x together withL_J-equivariant maps

Z^v_x −→

(

(l₁, . . . ,l_r)∈(L_J)^r

¯¯

¯

l⁻¹_i l_i+1∈¡

LJ∩PK_xi¢v_i¡

LJ∩PK_xi+1¢ l⁻¹_r ^Fl₁∈¡

LJ∩PK_xr¢ vrF¡

LJ∩PK_x₁¢ )

(2.6) with fibers isomorphic to

UJ∩¡x_rPI_rw_r^FPI₁Fx⁻¹₁ v⁻¹_r ¢

×

r−1Y

i=1

UJ∩¡x_iPI_iw_iPI_i+1x⁻¹_i+1v⁻¹_i ¢ . In the case where ^vⁱK_x_i+1 =K_x_i and ^v^r^FK_x₁ =K_x_r, the quotient by Q

LJ∩PK_xi

of the variety on the right-hand side of 2.6 can be identified with the parabolic Deligne-Lusztig variety X_L_J(Kx₁,v₁· · ·vrF).

2.3 A further decomposition

We now study the intersection (xP_Iw^FPIFx⁻¹v⁻¹)∩UJ in order to obtain information on the morphism Z^v_x−→XL_J(Kx,vF) defined at the end of Section 2.1. This will be achieved using the Curtis-Deodhar decomposition.

Let x,w,w^′ be elements ofW, and fix a reduced expression w=s₁· · ·sr of w.

Recall that a subexpressionofw(with respect to the decomposition w=s₁· · ·s_r) is an element of Γ⁼^{1,^s1}× · · · ×{1,s_r}. Such a subexpression γ=(γ₁, . . .,γr) is said to be x-distinguished if γi = si whenever xγ₁· · ·γi−1si > xγ₁· · ·γi−1. The main result in [7] and [5] gives a decomposition of the double Schubert cell BwB∩(B)^xw^′B⊂G/B in terms of certain x-distinguished subexpressions of w, as well as an explicit parametrisation of each piece (see [12, Section 2.2] for more details).

Theorem 2.7 (Deodhar, Curtis). Let w,w^′,x be elements of the Weyl group and w=s₁· · ·sr be a reduced expression ofw. There exists a decomposition ofBwB∩ (B)^xw^′Binto locally closed subvarieties