A note on decomposition numbers for groups of Lie type of small rank

A note on decomposition numbers for groups of Lie type of small rank

Olivier Dudas ^∗ December 21, 2012

Abstract

Using virtual projective characters coming from the ℓ -adic cohomology of Deligne-Lusztig varieties, we determine new decomposition numbers for principal blocks of some groups of Lie type of small rank, including G

(q) and Steinberg’s triality groups

D

(q).

Introduction

We are interested in the representation theory of finite reductive groups G(q) in non-defining characteristic ℓ ∤ q. As in the ordinary case, representations fall into modular Harish-Chandra series [9]. Each simple G(q)-module appears in the socle (and the head) of the parabolic induction of some cuspidal representa- tion, providing an inductive approach for studying the modular representation theory of G(q).

In this note we address the problem of finding these cuspidal representa- tions and their projective covers. For that purpose, we study the virtual projec- tive module P w coming from the mod-ℓ cohomology of a Deligne-Lusztig variety Y

(w). For elements w that are minimal in their conjugacy class, the results in [1] give some control on which projective indecomposable module can appear in P w and on its multiplicity. When the rank of G(q) is small, it turns out that this gives enough information to compute the character of projective covers of cuspidal modules.

We illustrate this method on some groups of F _q -rank 2, namely G ₂ (q) and Steinberg’s triality groups ³ D ₄ (q). For these groups, the decomposition matrices of the unipotent ℓ-blocks are known up to a few missing entries by [11] and [8].

Determining these entries using only algebraic methods seems quite challenging (see [13, 14] for Sp ₄ (q) and SU ₃ (q)), but turns out to be fairly elementary using the virtual modules P _w ’s.

∗

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

1

1 DELIGNE-LUSZTIG CHARACTERS 2

1 Deligne-Lusztig characters

1.1 Deligne-Lusztig varieties

Let G be a reductive algebraic group, together with an isogeny F , some power of which is a Frobenius endomorphism. In other words, there exists a positive integer δ such that F

defines a split F _q

-structure on G for a certain power q

of the characteristic p, where q ∈ R

. Let T ⊂ B be a maximal torus contained in a Borel subgroup B, both of which are assumed to be F -stable. The corresponding Coxeter system will be denoted by (W , S).

We choose a set of representatives { ˙ w} of W in N

(T) satisfying ( ˙ ww

) = w ˙ w ˙

whenever ℓ(ww

) = ℓ(w) + ℓ(w

). The Deligne-Lusztig varieties associated to w ∈ W are defined by

Y

( ˙ w) = ©

g ∈ G/U ¯ ¯ g

F (g) ∈ U wU ˙ ª

X

(w) = ©

g ∈ G/B ¯ ¯ g

F(g) ∈ BwB ª

πw

/

where π w denotes the restriction to Y

( ˙ w) of the canonical projection G/U −→

G/B. The varieties Y

( ˙ w) and X

(w) are quasi-projective varieties and endowed with a left action of G ^F by left multiplication. Furthermore, T ^{wF ˙} acts on the right on Y

( ˙ w) by right multiplication and π w is isomorphic to the correspond- ing quotient map, so that it induces a G ^F -equivariant isomorphism of varieties Y

( ˙ w)/T ^{wF ˙} → e X

(w).

1.2 Virtual characters

Let ℓ be a prime number different from p. We fix a finite extension K of Q

and we denote by O the ring of integers over Z

. It is a local ring and its residue field k is a finite field of characteristic ℓ. Throughout this paper, we assume that K is large enough so that K G ^F is a split semisimple algebra.

For a finite group H, we will denote by R _K (H) (resp. R _k (H), resp. P _k (H)) the Grothendieck group of the category of finitely generated K H-modules (resp. kH- modules, resp. projective kH-modules). We denote by 〈−,−〉 : R k (H)×P k (H) −→ Z the usual pairing, by d : R _K (H) −→ R _k (H) the decomposition map and by e : P _k (H) −→ R _K (H) its adjoint.

For w ∈ W and Λ ^{= K or} k, the cohomology groups H ⁱ _c (Y

( ˙ w), Λ ) with compact support are ( Λ ^{G F ,} Λ ^{T wF ˙} )-bimodules and their alternating sum yields functions R _w and R _w between the Grothendieck groups of T ^{wF ˙} and G ^F . By the universal coefficient formula, the following diagram commutes

R K (T ^{wF ˙} ) ^R

^{/ /}

d

R K (G ^F )

d

R _k (T ^{wF ˙} ) ^R

^{/ /} R _k (G ^F )

1 DELIGNE-LUSZTIG CHARACTERS 3 Furthermore, as we will see in the next section, the restriction of R _w induces a function P _k (T ^{wF ˙} ) −→ P _k (G ^F ) and therefore gives, for various w ∈ W , many virtual projective characters.

Note that the virtual characters R w (1) and R w (1) (which encode the coho- mology of X

(w)) involve only unipotent characters. They can be explicitely computed using only almost characters and Fourier matrices, and have been implemented in the package CHEVIE _of GAP3 _[10].

The fact that Deligne-Lusztig induction commutes with the decomposition map has the following consequence, which is often used to find lower bounds on decomposition numbers. Let θ be an ℓ-character, which we assume to be in general position. Then by [5, Corollary 9.9] (−1)

R _w (θ) is a proper character.

Moreover, its ℓ-reduction is d(R _w (θ)) = R _w (d(θ)) = R _w (1), and we deduce the following lemma.

Lemma 1.1. Assume there exists an ℓ-character of T ^{wF ˙} in general position. Then R _w (1) is a non-negative combination of simple kG ^F -modules.

We shall not give any detail for the existence of ℓ-characters in general po- sition since it is usually a straightforward computation. They correspond to ℓ-elements in the complement of the reflecting hyperplanes of C _W (wF) acting on X (T)/(wF − 1)X(T). In particular, they always exist when w is a d-regular element and ℓ | Φ d (q) is large enough.

1.3 Cohomology complexes and their character

In the modular setting it is often useful to work with a complex represent- ing the cohomology of a variety rather than the indivual cohomology groups.

Rickard has defined in [15] a representative for this complex with good finite- ness properties (see also [16]).

Proposition 1.2. There exists a complex R Γ ^{c (Y}

( ˙ w), k) of (kG ^F , kT ^{wF ˙} )-bimodules representing the cohomology of Y

( ˙ w) such that Res

R Γ c (Y

( ˙ w), k) is a bounded complex of finitely generated projective kG ^F -modules.

As a consequence, for any projective kT ^{wF ˙} -module Q, the terms of the com- plex R Γ c (Y

( ˙ w), k) ⊗ _kT

Q are projective kG ^F -modules and therefore its char- acter R _w ([Q]) is a virtual projective module. In particular, R _w (1) is a virtual projective character whenever ℓ ∤ |T ^{wF ˙} |.

Notation 1.3. We will denote by P _w ∈ P _k (G ^F ) the virtual module R _w ¡

[kT ^{wF ˙} ] ¢

.

Remark 1.4. Note that when ℓ | |T ^{wF ˙} |, the character R _w (1) is no longer pro-

jective. Nevertheless it differs from a virtual projective character by a linear

combination of non-unipotent characters. For unipotent blocks, the unipotent

characters are expected to form a basic set, so working with R _w (1) instead of P _w

does not make a big difference from our perspective.

1 DELIGNE-LUSZTIG CHARACTERS 4 The results in [1, §8] give information on how it is decomposed on the basis of projective characters. For N a simple kG ^F -module, we denote by P N the pro- jective cover of N. Then there exists w ∈ W such that ­

[N], P w ®

6= 0. Denote by W (N) the set of minimal elements for the Bruhat order such that this condition is satisfied. Then by [1, Propositions 8.10 and 8.12] the complex R Γ c (Y

( ˙ w), k) can be represented by a complex 0 −→ Q

−→ Q

−→ · · · −→ Q 2ℓ(w) −→ 0 of finitely generated kG ^F -modules such that

• for i > ℓ(w) and for any indecomposable summand P _N of Q _i there exists v ∈ W (N) such that v < w;

• for any indecomposable summand P _N of Q

there exists v ∈ W (N) such that v ≤ w.

This has the following consequence on the virtual character.

Proposition 1.5. Let N be a simple kG ^F -module, and w be a element of minimal length such that ­

[N], P _w ®

6= 0. Then there exists a > 0 such that (−1)

P _w = a[P _N ]+ X

[M]6=[N]

a _M [P _M ].

Furthermore, if a M < 0 then there exists v < w such that ­

[M],P v ® 6= 0.

Remark 1.6. Let I be a proper subset of S, and denote by W _I (resp. L I ) the parabolic subgroup of W (resp. the standard Levi subgroup of G). Then for w ∈ W _I , the complex R Γ c (Y

( ˙ w), k) is the Harish-Chandra induction of R Γ c (Y

( ˙ w), k).

Consequently, W (N) contains an element lying in a proper F-stable parabolic subgroup if and only if N is not cuspidal.

Remark 1.7. It turns out that in general W (N) are the elements of minimal length of a single F-conjugacy class.

1.4 Eigenspaces of the Frobenius

In some cases P w has too many consituents and we want to break it in smaller pieces (see the proof of Theorem 2.5). One way to do so is to consider eigenspaces on the complex of endomorphisms acting on the variety. Let σ be an endomorphism of Y

( ˙ w) acting on the right by equivalences of the étale site and by automorphisms of T ^{wF ˙} . We assume that the action on Y

( ˙ w) is compatible with the action of T ^{wF ˙} and commutes with the action of G ^F . Then Proposi- tion 1.2 can be generalised and R Γ c (Y

( ˙ w), k) can be endowed with an action of G ^F × (T ^{wF ˙} ⋊ ^〈 σ 〉 ) ^opp such that Res

R Γ c (Y

( ˙ w), k) is a bounded complex of finitely generated projective kG ^F -modules (see [7, §1.2]). In particular, for λ ∈ k, the generalised λ-eigenspace of σ

R Γ c (Y

( ˙ w), k)

: = R Γ c (Y

( ˙ w), k) ⊗ _k〈σ〉  k[σ] _(σ−λ)

is a complex of projective kG ^F -modules, and its character yields an element of

P k (G ^F ) which is in general smaller than P w . This character, or rather its im-

age by e can be computed from the various generalised µ-eigenspaces of σ on

2 EXAMPLES 5 H

_c (Y

( ˙ w),K ) when µ runs over the eigenvalues of σ whose ℓ-reduction is equal to λ.

Example 1.8. When G = SL ₂ with its split F q -structure, the eigenvalues of the Frobenius on H

_c (X(s), K ) are 1 and q. If ℓ ∤ q − 1, the ℓ-reduction of these eigenvalues remain distinct. In that case, the generalised 1-eigenspace and q- eigenspace of F on H

_c (Y(s), k) yields two virtual projective modules. They lift to projective characters that have only one unipotent constituent, namely the Steinberg character when λ = 1 and the trivial character when λ = q.

A typical example of endomorphism acting on Y

( ˙ w) is the Frobenius F

. By a result of Lusztig, the eigenvalues of F

on a given unipotent representation ρ in the cohomology of X

(w) are of the form λ

q ^sδ/2 where λ

is a root of unity depending only on ρ and s is a non-negative integer. The integer s can be com- puted using the formula in [6, Corollaire 3.3.8], which gives, for m sufficiently divisible, the trace of F ^m on the cohomology of X

(w). When (G,F ) is assumed to be split, or equivalently when δ = 1, it can be written

X ( − 1) ⁱ Tr ¡

F ^m , H ⁱ _c (X

(w),K ) ¢

= X

χ∈IrrK

W

R

· χ q

(T _w )

where R

is the almost character associated with χ and χ q

is the character of the Hecke algebra H _q

(W ) which specializes in χ for q = 1.

Remark 1.9. When F ( ˙ w) = w ˙ then not only F

but F acts on Y

( ˙ w) and one can compute its eigenvalues using Lefschetz trace formula (when (G, F ) is not a Ree or Suzuki group) and the fact that Y

( ˙ w) ^F = ∅ when w 6= 1. Other natural endomorphisms of X

(w) have been defined in [2] but one does not know if they always satisfy Lefschetz trace formula.

2 Examples

Let (G, F) be a finite reductive group of small rank, and b be the idempo- tent associated with the principal ℓ-block. We consider, for each cuspidal simple kG ^F -module N in the principal block, the virtual character (−1)

bR w (1) for w ∈ W (N) (see §1.3 for the notation). Up to adding and removing non-unipotent characters, it corresponds to the virtual projective module (−1)

P w whose co- efficient on [P N ] is positive, giving in some particular cases enough information to determine e([P _N ]).

We shall denote the unipotent characters as in CHEVIE . Note that it slightly differs from the one in Carter’s book [3, §13].

2.1 The group G ₂ (q)

Let (G, F) be a split group of type G ₂ . We denote by s and t the two simple

reflections of the associated Weyl group. When ℓ is a good prime number the

2 EXAMPLES 6 decomposition matrix of the ℓ-blocks have been determined in [11] with the ex- ception of a few decomposition numbers when ℓ | q +1. We shall determine these numbers for the principal ℓ-block.

Theorem 2.1. Let ℓ | q + 1 be a good prime. Let ℓ ^r be the highest power of ℓ dividing q + 1. If ℓ ^r 6= 5, then the unipotent part of the decomposition matrix of the principal ℓ-block is given by

φ _1,0 1 · · · · · φ

_1,3 1 1 · · · · φ

_1,3 1 · 1 · · · G ₂ [1] · · · 1 · · G ₂ [−1] · · · · 1 · φ _1,6 1 1 1 2 2 1

Proof. It is shown in [11, Theorem B] that the decomposition matrix of the prin- cipal ℓ-block has the following shape

P ₁ P ₂ P ₃ P ₄ P ₅ P ₆ φ 1,0 1 · · · · · φ

_1,3 1 1 · · · · φ

_1,3 1 · 1 · · · G ₂ [1] · · · 1 · · G ₂ [−1] · · · · 1 · φ _1,6 1 1 1 α β 1

with α, β ≥ 2 when ℓ ^r 6= 5 (this bound can be obtained by Lemma 1.1). Since the ℓ-reduction of the characters G ₂ [1] and G ₂ [−1] remains irreducible, one can easily check that W (G ₂ [−1]) = {st, ts} and W (G ₂ [1]) = {stst, tsts}. Indeed, these elements are the minimal elements w such that the ordinary characters appear as a constituent of R w (1).

The values of R w (1) for various elements w ≤ stst are given in the following table.

w (−1)

bR _w (1) (−1)

bP _w 1 φ _1,0 + φ

_1,3 + φ

_1,3 + φ _1,6 [P ₁ ]

s, tst − φ 1,0 − φ

_1,3 + φ

_1,3 + φ 1,6 −[P ₁ ]+ 2[P ₂ ] t, sts − φ 1,0 − φ

_1,3 + φ

_1,3 + φ 1,6 − [P ₁ ] + 2[P ₃ ] st, ts φ 1,0 + φ 1,6 + G ₂ [ − 1]

stst φ _1,0 + φ _1,6 + G ₂ [1]

Here, in the first three rows, we have decomposed bP _w on the basis of projec-

tive indecomposable characters using the unitriangularity of the decomposition

matrix. In particular, we observe that [P ₆ ] does not occurs in the cohomology of

Y

( ˙ w) for w < st (this is also explained by the fact that the head of P ₆ is actually

cuspidal). Therefore, there exists integers a ₁ ,a ₂ , a ₃ , a ₅ , a ₆ with a ₅ , a ₆ ≥ 0 such

that

2 EXAMPLES 7 X a _i e ¡

[P _i ] ¢

= φ 1,0 + φ 1,6 +G ₂ [ − 1] + non-unip.

Using the unitriangularity of the decomposition matrix we deduce that a ₁ = 1, a ₂ = a ₃ = −1, a ₅ = 1 and therefore by projection on the unipotent characters we obtain the relation ( − 1 + β + a ₆ )φ _1,6 = φ _1,6 , which forces a ₆ = 2 − β. Since β ≥ 2 and a ₆ ≥ 0 we obtain β = 2 and a ₆ = 0. As a consequence, [P ₆ ] does not occurs in the cohomology of Y

( ˙ w) for w < stst.

Finally, we use the same argument for decomposing P _w for w = stst as [P ₁ ] − [P ₂ ] − [P ₃ ] + [P ₄ ] and we deduce that α = 2.

Remark 2.2. The assumption on ℓ ^r is necessary. Indeed, when q = 2 ⁿ and ℓ ^r = 5, then it is shown in [17] that β = 2 and α = 1.

An interesting feature of this method is that it can be applied to construct decomposition matrices of spetses, as in [4]. Let (G, F) be a spets of type I ₂ (2m).

Then G ^F has order (q ^2m − 1)(q ² − 1) so that the principal (q + 1)-block has non- cyclic generic defect. The unipotent characters in the principal block are the trivial character φ _1,0 , the Steinberg character φ 1,2m , two character φ

_1,m and φ

_1,m corresponding to the two one-dimensional characters of the dihedral group D _2m and m− 1 cuspidal characters denoted by I ₂ [ j, m] for j = 1, . . ., m− 1. If we assume that Howlett-Lehrer theory extend to spetses we obtain the first three columns by Harish-Chandra induction from a split torus and the two Levi subgroups of rank 1. The last column is the analogue of the Gelfand-Graev module. Moreover, if we believe in Geck’s conjecture on the shape of the decomposition matrix (see [9, Conjecture 3.4]) we get a matrix of the form

P ₁ P ₂ P ₃ Q ₁ Q ₂ · · · Q m−1 P ₄

φ 1,0 1 · · · · · · ·

φ

_1,m 1 1 · · · · · · φ

_1,m 1 · 1 · · · · · I ₂ [m − 1, m] · · · 1 · · · · I ₂ [m − 2, m] · · · · 1 · · ·

.. . · · · · · . .. · ·

I ₂ [1, m] · · · · · · 1 ·

φ 1,6 1 1 1 α 1 α 2 · · · α m−1 1

Note that if we can make sense of tensor products φ 2, j ⊗I ₂ [1, j] one should obtain a projective character of the form [Q j ] + f (q)[P ₄ ] for some polynomial f . Fur- thermore, since we are working in the generic setting, an analogue of Lemma 1.1 should apply and would give α j ≥ 2 for all j.

Finally, and most importantly, the virtual character R w (1) makes sense for a spets of type I ₂ (2m) since the Fourier matrix for the family containing the cuspidal characters is explicitely known by [12]. We find

and

bR _(st)

(1) = φ 1,0 + φ 1,2m + I ₂ [ j, m]

bR _s (1) = φ _1,0 + φ

_1,m − φ

_1,m − φ 1,2m

bR _t (1) = φ _1,0 + φ

_1,m − φ

_1,m − φ 1,2m

2 EXAMPLES 8 so that we can apply the same argument as in the proof of Theorem 2.1 to obtain α j = 2 for all j = 1, . . ., m − 1.

2.2 Steinberg’s triality groups ³ D ₄ (q)

Let (G, F) be a group of type ³ D ₄ . We denote by s, t, u, v the simple reflections of the associated Weyl group with the convention that s, t, u commute. When ℓ is a good prime number the decomposition matrix of the ℓ-blocks have been investigated by Geck in [8] using modified generalised Gelfand-Graev represen- tations. Some entries in the decomposition matrices of the principal Φ d -blocks for d = 2, 3, 6 are still unknown. They correspond to projective covers of cuspidal modules. The purpose of these section is to determine some of these entries.

Theorem 2.3. Let ℓ 6= 2, 3 be a prime dividing q + 1. Let ℓ ^r be the highest power of ℓ dividing q + 1. If ℓ ^r 6= 5, then the unipotent part of the decomposition matrix of the principal ℓ-block is given by

φ _1,0 1 · · · · · φ

_1,3 1 1 · · · · φ

_1,3 1 · 1 · · ·

3 D ₄ [1] · · · 1 · ·

3 D ₄ [−1] · · · · 1 ·

φ 1,6 1 1 1 2 2 1

Proof. By [8], there exists positive integers α,β such that the decomposition matrix of the principal ℓ-block is

P ₁ P ₂ P ₃ P ₄ P ₅ P ₆ φ 1,0 1 · · · · · φ

_1,3 1 1 · · · · φ

_1,3 1 · 1 · · ·

3 D ₄ [1] · · · 1 · ·

3 D ₄ [−1] · · · · 1 ·

φ 1,6 1 1 1 α β 1

When ℓ ^r 6= 5, the integers α and β are actually larger than 2. Indeed, one checks that T ^{w ˙}

^F contains a subgroup isomorphic to a Φ 2 -torus of a group of type G ₂ , therefore it contains an ℓ-character in general position whenever ℓ ^r 6= 5 (note that C W (w ₀ F) ≃ W ^F is isomorphic to a Weyl group of type G ₂ ). Using Lemma 1.1 we get the lower bound for α and β.

Now we can proceed as in the proof of 2.1, and compute the different projec-

tive characters coming from the Deligne-Lusztig varieties. We obtain

2 EXAMPLES 9

w (−1)

bR w (1) (−1)

bP w

1 φ 1,0 + φ

_1,3 + φ

_1,3 + φ 1,6 [P ₁ ]

s, u, vuv, svu, suv − φ _1,0 − φ

_1,3 + φ

_1,3 + φ _1,6 − [P ₁ ] + 2[P ₂ ] v − φ _1,0 − φ

_1,3 + φ

_1,3 + φ _1,6 − [P ₁ ] + 2[P ₃ ] sv, uv φ _1,0 + φ _1,6 + ³ D ₄ [−1]

svuv φ _1,0 + φ _1,6 + ³ D ₄ [1] + 2 ¡

φ

_1,3 + φ

_1,3 − ³ D ₄ [−1] ¢

Using the unitriangularity of the decomposition matrix and the fact that α,β ≥ 2, we deduce that [P ₆ ] does not appear in the cohomology of varieties Y

(w) for w ≤ svtv and we find

and

bP sv = [P ₁ ]− [P ₂ ]− [P ₃ ] + [P ₅ ] bP _svuv = [P ₁ ] + [P ₂ ] + [P ₃ ] + [P ₄ ] − [P ₅ ] so that we have necessarily α = β = 2.

For the principal Φ 3 -block, the method in §1.3 gives only partial information on the decomposition matrix.

Theorem 2.4. Let ℓ 6= 2, 3 be a prime dividing q ² + q + 1. Then the unipotent part of the decomposition matrix of the principal ℓ-block is given by

φ 1,0 1 · · · · · · φ

_1,3 · 1 · · · · · φ 2,1 · 1 1 · · · · φ 2,2 1 · · 1 · · ·

3 D ₄ [1] · · · · 1 · ·

φ

_1,3 · · 1 · a 1 · φ _1,6 · · · 1 2a − 3 2 1 where a ≥ 2.

Proof. By [8], the unipotent part of the decomposition matrix is given by P ₁ P ₂ P ₃ P ₄ P ₅ P ₆ P ₇

φ 1,0 1 · · · · · · φ

_1,3 · 1 · · · · · φ _2,1 · 1 1 · · · · φ 2,2 1 · · 1 · · ·

3 D ₄ [1] · · · · 1 · ·

φ

_1,3 · · 1 · a 1 · φ _1,6 · · · 1 b c 1

where a ≥ 1 and c ≥ 2. The cohomology of X

(sv) gives c = 2, and one can check by looking at the cohomology of various Y

( ˙ w) that [P ₇ ] does not occur for w <

stuv.

2 EXAMPLES 10 If ℓ 6= 2, 3, one can check that there always exists an ℓ-character of T ^{wF ˙} in general position with w = (stuv) ² . By Lemma 1.1 this forces 3 − 2a + b ≥ 0. Now, the cohomology of X

(stuv) is given by

bR _w (1) = φ _1,0 + φ _1,6 − 3φ _2,2 + 2φ

_1,3 + 2φ

_1,3 + ³ D ₄ [1]

= e ¡

[P ₁ ] + 2[P ₂ ] − 2[P ₃ ] − 4[P ₄ ]+ [P ₅ ] + m[P ₆ ] + n[P ₇ ] ¢

+ non-unip.

with n ≥ 0. By comparing the two characters we get a = 4 − m and b = 5 −2m − n.

But then 3 − 2a + b = −n is non-negative and non-positive so we must have n = 0 and therefore b = 2a − 3.

The last block illustrates the importance to consider eigenspaces of the Frobe- nius on the cohomology complex.

Theorem 2.5. Let ℓ 6= 2, 3 be a prime dividing q ² − q + 1. Then the unipotent part of the decomposition matrix of the principal ℓ-block is given by

φ _1,0 1 · · · · · · φ

_1,3 · 1 · · · · · φ _2,2 1 1 1 · · · ·

3 D ₄ [ − 1] · · · 1 · · ·

3 D ₄ [1] · · · · 1 · ·

φ

_1,3 1 · 1 · · 1 · φ 1,6 · 1 1 2 c · 1 where c = 0 or 1.

Proof. By [8], the unipotent part of the decomposition matrix is given by P ₁ P ₂ P ₃ P ₄ P ₅ P ₆ P ₇

φ 1,0 1 · · · · · · φ

_1,3 · 1 · · · · · φ 2,2 1 1 1 · · · ·

3 D ₄ [−1] · · · 1 · · ·

3 D ₄ [1] · · · · 1 · ·

φ

_1,3 1 · 1 a · 1 · φ _1,6 · 1 1 b c d 1

where 2a+b −2 ≥ 0. The projective modules P i for i = 4, . . ., 7 are cuspidal so they do not occur in the comology complex of the varieties Y

( ˙ w) for w ∈ {1, s, t, u, v}.

The character of the variety associated with sv, together with the unitriangular- ity of the decomposition matrix, gives

e ¡

[P ₁ ]− [P ₃ ]+ [P ₄ ]+ m[P ₆ ] + n[P ₇ ] ¢

= φ 1,0 + φ 1,6 + ³ D ₄ [−1] + non-unip.

with m, n ≥ 0. Looking at the coefficients on φ

_1,3 and φ _1,6 , we obtain the relations a + m = 0 and − 1 + b + md + n = 1. This forces a = m = 0 and b + n = 2. Since 2a + b − 2 ≥ 0, we obtain b = 2.

If we consider now the virtual character associated with Y

( ˙ w) for w = stuv, we obtain a relation for [P ₅ ] and [P ₆ ] given by e ¡ [P ₅ ] + m[P ₆ ]+ n[P ₇ ] ¢

= ³ D ₄ [1] +

REFERENCES 11 7φ

_1,3 + 9φ 1,6 . This gives bounds for c and d that are not really satisfactory. To get smaller bounds, one can consider direct summands of the complex coming from eigenspaces of the Frobenius. Here, not only F

= F ³ acts on Y

( ˙ w) but also F since w is F -stable. The eigenvalues of F can be deduced from the eigenvalues of F ³ together with the fact that Y

( ˙ w) ^F = ∅ (see Remark 1.9). The characters of the eigenspaces are given by

i £

bR Γ c (X

(w),K ) _(q

)

¤ 0 φ 1,6 − φ 2,2

1 φ

_1,3 − ³ D ₄ [−1]

2 D ₄ [1] − φ _2,2 3 φ

_1,3 − ³ D ₄ [−1]

4 φ 1,0 − φ 2,2

5 φ

_1,3 + φ

_1,3

They are all the unipotent part of virtual projective characters which decom- pose on the basis of the e([P _i ]) with non-negative coefficients for i = 5, 6, 7. With e([P ₂ ] − [P ₃ ]) = φ

_1,3 − φ

_1,3 + non-unip, we deduce from the last eigenspace that d = 0. Using the generalised (q ² )-eigenspace and the virtual projective charac- ter −[P ₃ ], we get c ≤ 1.

Remark 2.6. According to the geometric version of Broué’s conjecture the action of F on R Γ c (X

(stuv), K ) should extend to an action of a 6-cyclotomic Hecke alge- bra H associated with the complex reflection group C _W (wF) of type G ₄ and with parameters [1; q ² ; −q]. A consequence of the conjectures in [2] is that the image of each projective k H -module by R Γ c (X

(stuv), k) ⊗ _k H − lifts to an irreducible character of K G ^F . This would show that c = 0.

Acknowledgements

This work was carried out during my visit at the RWTH Aachen Univer- sity. I would like to thank Gerhard Hiss and Frank Lübeck for their invitation, their support and many valuable and stimulating discussions which started this project.

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