Olivier Dudas

Cohomology of Deligne-Lusztig varieties for short-length regular elements in exceptional groups

We determine the cohomology of Deligne-Lusztig varieties associated to some short-length regular elements for split groups of type F 4 and E n . As a byproduct, we obtain conjectural Brauer trees for the principal Φ 14 -block of E 7 and the principal Φ 24 -block of E 8 .

Introduction

Let G be a finite group and ℓ be a prime number. The ℓ-modular representation theory of G is somehow controlled by the representation theory of local subgroups, namely the ℓ-subgroups of G and their normalisers. Broué's abelian defect conjecture is one of the major open problems in this framework: it predicts that an ℓ-block of G with abelian defect group is derived equivalent to its Brauer correspondent. From the work of Rickard [START_REF] Rickard | Morita theory for derived categories[END_REF], we know that such an equivalence should be induced by a perfect complex. Unfortunately, there is no canonical construction in general.

When G = G F is a finite reductive group, Broué's suggested that the complex representing the cohomology of some Deligne-Lusztig variety should be a good candidate. Together with Michel in [START_REF] Broué | Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées[END_REF], they made precise which specific Deligne-Lusztig varieties would be associated to principal Φ d -blocks when d is a regular number. They introduced the notion of good d-regular elements w ∈ W and conjectured that • for i = j, the groups H i c (X(w), Q ℓ ) and H j c (X(w), Q ℓ ) have no irreducible constituents in common;

• the irreducible constituents of H • c (X(w), Q ℓ ) are exactly the unipotent characters lying in the principal Φ d -block;

• the endomorphism algebra End G F H • c (X(w), Q ℓ ) is a d-cyclotomic Hecke algebra.

As for now, these statements have been veryfied in very few cases only. Computing the cohomology of a Deligne-Lusztig variety is a difficult problem, and the only results in this direction have been obtained by Lusztig in [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] when d is the Coxeter number (that is when w is a Coxeter element of W), for groups of rank 2 by Digne, Michel and Rouquier in [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF] and for d = n in type A n and d = 4 in type D 4 by Digne and Michel in [START_REF] Digne | Endomorphisms of Deligne-Lusztig varieties[END_REF]. The purpose of this paper is to provide new examples for exceptional groups and in the spirit of Broué's conjecture, to deduce structural properties of the corresponding Φ d -block.

We shall adapt Lusztig's strategy: if a character is non-cuspidal then it should appear in the cohomology of a certain quotient of the Deligne-Lusztig variety X(w). In the Coxeter case, Lusztig proved that this quotient can be expressed in terms of a Deligne-Lusztig variety associated to a "smaller" Coxeter element, providing an inductive method to compute the cohomology of X(w). The first step towards our main result is to show an analogous property for the dregular elements we are interested in. To this end we will make extensive use of [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF]. Unfortunately, this will not give enough information to deal with nonprincipal series. In order to compute the cuspidal part of the cohomology of X(w), we shall, as in [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF], introduce compactifications of X(w). Unlike the Coxeter case, the cuspidal part of H • c (X(w), Q ℓ ) might not be concentrated in degree ℓ(w) since some of the divisors of X(w) might provide cuspidal characters. However, the results in [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF] are sufficient to determine explicitely in which groups they actually appear and we obtain the following result:

Theorem. Let w be a good d-regular element. Then the contribution of the principal series and the discrete series to the cohomology of the Deligne-Lusztig variety X(w) is explicitely known in the following cases:

• (G, F) has type F 4 and d = 8;

• (G, F) has type E 6 and d = 9;

• (G, F) has type E 7 and d = 14;

• (G, F) has type E 8 and d = 24.

We will also obtain partial results for the other series, as well as predictions on their contribution, in line with the formula given by Craven in [START_REF] Craven | On the cohomology of Deligne-Lusztig varieties[END_REF]. Using Lusztig's results in the Coxeter case, Hiss, Lübeck and Malle have conjectured that the Brauer tree of the principal Φ h -block can be read off the cohomology of the Coxeter variety [START_REF] Hiss | The Brauer trees of the exceptional Chevalley groups of type E 6[END_REF]. Using the existing Brauer trees and the previous theorem, we propose conjectural planar embedded Brauer trees for the principal Φ 14 -block of E 7 and for the principal Φ 24 -block of E 8 (see Figure 3 and4). We believe that a further study of the cohomology of the corresponding Deligne-Lusztig varieties as in [START_REF] Dudas | Coxeter Orbits and Brauer trees I[END_REF], [START_REF] Dudas | Coxeter Orbits and Brauer trees II[END_REF] and [START_REF] Dudas | Coxeter Orbits and Brauer trees III. in preparation[END_REF] should give credit to these predictions.

Methods for determining the cohomology

Let G be a connected reductive group, together with a Frobenius F defining a F q -structure on G. If H is any F-stable algebraic subgroup of G, we will denote by H the finite group of fixed points H F . We fix a Borel subgroup B containing a maximal torus T of G such that both B and T are F-stable. The associated Weyl group is W = N G (T)/T and the set of simple reflections will be denoted by S. We will assume that (G, F) is split, so that F acts trivially on W.

Recall from [START_REF] Deligne | Representations of reductive groups over finite fields[END_REF] that to any element w ∈ W one can associate a Deligne-Lusztig variety

X(w) = gB ∈ G/B | g -1 F g ∈ BwB .
It is a quasi-projective variety of pure dimension ℓ(w), on which G acts by left multiplication. This definition has been subsequently generalized in [START_REF] Broué | Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées[END_REF] to elements of the Artin-Tits monoid B + .

The ℓ-adic cohomology of X(w) carries a lot of information on ordinary and modular representations of G. Throughout this paper, we will be interested in the case where w is a good d-regular element, or equivalently when w is a droot of π = w 2 0 in the Braid group of W. In that case, it is conjectured that the cohomology of X(w) gives a good model for the unipotent part of the principal Φ d -block (see for example [START_REF] Broué | Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées[END_REF] and [START_REF] Broué | Generic blocks of finite reductive groups[END_REF] or the introduction for more details).

Non-cuspidal characters

To any subset I ⊂ S one can associate a standard parabolic subgroup P I containing B and a standard Levi subgroup L I containing T. If U I denotes the unipotent radical of P I , the parabolic subgroup decomposes as P I = L I U I and both L I and U I are F-stable. By [17, XVII, 6.2.5], the U I -invariant part of the cohomology of X(w) is isomorphic to the cohomology of U I \X(w). Consequently, one can detect the presence of non-cuspidal modules in the cohomology of X(w) by studying the quotient variety U I \X(w) for various subsets I. In some specific cases, we can express such a quotient (or at least its cohomology) by means of smaller Deligne-Lusztig varieties.

Let b = w 1 • • • w r ∈ B + be an element of the Braid monoid, written as a product of reduced elements (i.e. w i ∈ W). Recall from [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF] that the decomposition of G/B into P I -orbits induces a decomposition of X(b) into locally closed P I -subvarieties, called pieces

X (W I x 1 ,...,W I x r ) (b) = X(b) ∩ P I x 1 B/B × • • • × P I x r B/B
where each x i runs over the set of I-reduced elements of W. When I and b are clear from context, we shall simply denote this variety by X (x 1 ,...,x r ) . Throughout this paper, we will make extensive use of a particular case of the main theorem of [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF]. It can be deduced from [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF]Remark 3.12] when each set I i is empty and all the x i 's are equal to the same element:

Theorem 1.1. Let b = w 1 • • • w r ∈ B +
with w i ∈ W, I be a subset of S and x be an I-reduced element of W. We assume that each w i can be decomposed as

w i = γ i w ′ i with γ i ∈ S ∪ {1} and w ′ i ≤ w i be such that (a) if γ i = 1 then v i = xw i x -1 ∈ W I ; (b) if γ i ∈ S then xγ i x -1 ∉ W I , v i = xw ′ i x -1 ∈ W I and ℓ(w ′ i ) = ℓ(v i ).
Let d be the number of w i 's satisfying condition (b) and e = dim(U x I ∩ w ′ i U∩U -). Then we have the following isomorphism of graded L I × 〈F〉-modules:

H • c (U I \X (x,...,x) ) ≃ H • c (G m ) d × X L I (v 1 • • • v r ) [-2e](e).
Remark 1.2. In the particular cases we will be interested in, b will always be reduced. In that case, it corresponds to an element w ∈ W and we have

w = w 1 • • • w r with ℓ(w) = ℓ(w 1 ) + • • • + ℓ(w r ).
Note that in general the variety X(b) ⊂ (G/B) r can have much more pieces that X(w) ⊂ G/B, since

X W I x (w) = x 2 ,...,x r I-reduced X (W I x,W I x 2 ,...,W I x r ) (b).
However, in our specific examples we will observe that the piece X (W I x,W I x 2 ,...,W I x r ) will be empty unless

x 2 = • • • = x r = x, so that X x ≃ X (x,x,...,x) .

Cuspidal characters

By definition, cuspidal representations of G have no non-zero element invariant under the action of U I unless I = S. In particular, the cohomology of the quotient variety U I \X(w) contains no information on the cuspidal characters that can appear in X(w). In this section we shall briefly review some methods developped in [START_REF] Digne | Endomorphisms of Deligne-Lusztig varieties[END_REF] and [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF] in order to solve the problem of finding cuspidal characters in the cohomology of Deligne-Lusztig varieties.

Let b

= w 1 • • • w r with w i ∈ W. Recall that the variety X(b) has a nice com- pactification X(w 1 • • • w r ) = (g 0 , g 1 , . . ., g r ) ∈ (G/B) r+1 g -1
i-1 g i ∈ Bw i B and g -1 r F(g 0 ) ∈ B which has the following properties (see [START_REF] Digne | Endomorphisms of Deligne-Lusztig varieties[END_REF] and [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF]) : Proposition 1.3. Let w 1 , . . ., w r be elements of W,

(i) X(w 1 • • • w r ) is a projective variety of dimension ℓ(w 1 ) + • • • + ℓ(w r ); (ii) X(w 1 • • • w r ) is smooth whenever each variety Bw i B is; (iii) X(w 1 • • • w r ) is rationally smooth whenever each variety Bw i B is; (iv) X(w 1 • • • w r ) has a filtration by closed subvarieties X(v 1 • • • v r ) where the v i 's satifisfy v i ≤ w i .
Remark 1.4. A particular case is when each w i is a simple reflection s i . Then the variety X(w 1 • • • w r ) coincides with the smooth compactification introduced by Deligne and Lusztig in [START_REF] Deligne | Representations of reductive groups over finite fields[END_REF].

Let w ∈ W. In order to compute the cuspidal part of the cohomology of X(w) using the previous compactifications, we will use the following results: (C1) the cohomology of X(w) over Q ℓ is zero outside the degress ℓ(w), . . ., 2ℓ(w) [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF]Corollary 3.3.22];

(C2) the following triangle is distinguished in D b (Q ℓ G-Mod) : RΓ c X(w), Q ℓ -→ RΓ c X(w), Q ℓ -→ RΓ c v<w X(v), Q ℓ (C3)
of F on H ℓ(w) c (X(w), Q ℓ ) ρ satisfy |λ| < |q ℓ(w)/2 |
. This is a particular case of [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF]Proposition 3.3.21].

Note finally that the property of being rationally smooth of X(w) can be read off the Kazdhan-Lusztig polynomials of W [11, proposition 3.2.5]. If X(w) happens to be not rationally smooth, we can always decompose w into a product w = w 1 • • • w r such that each variety Bw i B is.

Some particular cases

For short-length regular elements, one can observe that only a small number of pieces X x are non-empty. In addition, they very often satisfy the assumptions of Theorem 1.1. For some of these elements, we can therefore compute explicitely the cohomology of the quotient U I \X(w), and eventually deduce the cohomology of X(w) using the results discussed in Section 1.2.

To make the computations easier, we shall use the notation introduced in [START_REF] Digne | Endomorphisms of Deligne-Lusztig varieties[END_REF]: the cohomology of the Deligne-Lusztig variety X(w) as a graded G × 〈F〉 monmodule will be represented by a polynomial H X(w) (t 1/2 , h) with coefficients in the semi-group N IrrG. By a theorem of Lusztig, when ρ is a unipotent character, any eigenvalue of F on the ρ-isotypic part of H i c (X(w), Q ℓ ) can be written λ ρ q j/2 , where λ ρ is a root of unity independent of w and i. The multiplicity of ρ in H i c (X(w), Q ℓ ) with eigenvalue λ ρ q j/2 will be encoded by the coefficient of h i t j/2 in the polynomial H X(w) (t 1/2 , h). For example, if X(s) is the Drinfeld curve for G = SL 2 (F p ) then H X(w) = hSt + h 2 tId.

Since we are dealing with exceptional Weyl groups, and more specifically with the combinatorics of distinguished subexpressions, we will use the package CHEVIE in GAP. We have written a couple of useful functions to determine whether a piece of a Deligne-Lusztig variety is non-empty, and to describe it that case its quotient by a finite unipotent group. These functions can be found in [START_REF] Michel | Development version of the GAP part of CHEVIE[END_REF] (or will be soon available) under the name deodhar.g.

8-regular elements for groups of type F 4

Let (G, F) be a split group of type F 4 . To fix the notation we will consider the following Dynkin diagram:

t 1 t 2 t 3 t 4 >
where t 1 , t 2 , t 3 and t 4 are the simple reflections.

Recall that there exist d-regular elements for d ∈ {1, 2, 3, 4, 6, 8, 12} only (see for example [START_REF] Springer | Regular elements of finite reflection groups[END_REF]). Note that the largest integer corresponds to the Coxeter number. By [START_REF] Broué | Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées[END_REF], for any such d one can find a particular d-regular element which is a d-th root of π in the Braid monoid. By [1, 11.22] and [10, Proposition 5.5], the cohomology of the corresponding Deligne-Lusztig variety does not depend on the choice of a root. For our purposes we will take

w = t 1 t 2 t 3 t 2 t 3 t 4 .

Cohomology of U I \X(w)

. We start by computing the cohomology of the quotient U I \X(w) where I = {t 2 , t 3 }. Using the criterion given in [START_REF] Digne | Endomorphisms of Deligne-Lusztig varieties[END_REF]Lemma 8.3] and the package CHEVIE in GAP one can check that there are only three non-empty pieces X x , corresponding to the cosets W I x = W I w 0 , W I w 0 t 1 t 2 and W I w 0 t 4 t 3 . Theorem 1.1 does not apply directly to all of these cells, but we can add an intermediate step. Let J = {t 2 , t 3 , t 4 } and K = {t 1 , t 2 , t 3 }. We have W J w 0 = W J w 0 t 4 t 3 (resp. W K w 0 = W K w 0 t 1 t 2 ) whereas X W I w 0 t 1 t 2 = X W J w 0 t 1 t 2 (resp. X W J w 0 t 4 t 3 ) is stable by P J (resp. P K ). Therefore only two pieces appear in the decompositon of U J \X(w) (resp. U K \X(w)).

• Let y be the minimal element of W J w 0 t 1 t 2 . Since t 1 t 2 is J-reduced, y = w J w 0 t 1 t 2 . Let us decompose w as w = w 1 w 2 with w 1 = t 1 t 2 t 3 t 2 and w 2 = t 3 t 4 = t 3 w ′ 2 . We have y w 1 = t 2 t 3 ∈ W J and y w ′ 2 = t 4 and therefore we can apply Theorem 1.1 to compute the cohomology of the piece of X(w 1 w 2 ) corresponding to (W J y,W J y). Furthermore, one can check (using GAP again) that the pieces corresponding to (W J y,W J y ′ ) are empty unless y and y ′ lie in the same coset. In particular, X W J y (w) ≃ X W J y,W J y (w 1 w 2 ) and

H • c (X W J y , Q ℓ ) U J ≃ H • c (G a × G m × X L J (t 2 t 3 t 4 ), Q ℓ ). Now X L J (t 2 t 3 t 4
) is a Deligne-Lusztig variety associated to a Coxeter element, and therefore the cohomology of its quotient by U I ∩ L J is given by [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF]Corollary 2.10]. We obtain

H • c (X W I w 0 t 1 t 2 , Q ℓ ) U I ≃ H • c (G a × G m × X L J (t 2 t 3 t 4 ), Q ℓ ) U I ∩L J ≃ H • c (G a × (G m ) 2 × X L I (t 2 t 3 ), Q ℓ ).
• For the piece X W I w 0 t 4 t 3 we proceed as above: the minimal element z of W K w 0 t 4 t 3 is clearly z = w K w 0 t 4 t 3 since t 4 t 3 is K -reduced. We can decompose w as w = w 1 w 2 w 3 where w 1 = t 1 , w 2 = t 2 = t 2 w ′ 2 and w 3 = t 3 t 2 t 3 t 4 . We observe that z w 1 = t 1 , z w ′ 2 = 1 and z w 3 = t 2 t 3 are elements of W K . In addition, we can check by explicit computation that a piece of X(w 1 w 2 w 3 ) corresponding to (W K z,W K z ′ ,W K z ′′ ) is empty unless z, z ′ and z ′′ lie in the same coset. Consequently, we can apply Theorem 1.1 to relate the cohomology of X W K z to the cohomology of X L K (t 1 t 2 t 3 ) and then use [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] to obtain

H • c (X W I w 0 t 4 t 3 , Q ℓ ) U I ≃ H • c (G a × G m × X L K (t 1 t 2 t 3 ), Q ℓ ) U I ∩L K ≃ H • c (G a × (G m ) 2 × X L I (t 2 t 3 ), Q ℓ ).
• For the open piece X W I w 0 we can directly apply Theorem 1.1 by decomposing w as w = w 1 w 2 with w 1 = t 1 (t 2 t 3 t 2 t 3 ) and w 2 = t 4 . We only have to check that X W I w 0 (w) = X (W I w 0 ,W I w 0 ) (w 1 w 2 ) which can be done using GAP.

We deduce

H • c (X W I w 0 , Q ℓ ) U I ≃ H • c ((G m ) 2 × X L I (t 2 t 3 t 2 t 3 ), Q ℓ ).
Note that the variety X W I w 0 t 1 t 2 ∪ X W I w 0 t 4 t 3 is closed in X(w). Furthermore, the elements in W I w 0 t 1 t 2 and W I w 0 t 4 t 3 are not comparable in the Bruhat order and therefore both X W I w 0 t 1 t 2 and X W I w 0 t 4 t 3 are closed subvarieties of the union.

In particular

H • c X W I w 0 t 1 t 2 ∪ X W I w 0 t 4 t 3 , Q ℓ U I ≃ H • c G a × (G m ) 2 × X L I (t 3 t 2 ), Q ℓ ⊕2 .
The Weyl group of L I has type B 2 . Let us denote by ε the sign representation of W I , by θ the one-dimensional representation such that θ(t 2 ) = 1 and θ(t 3 ) = -1 and by r the reflection representation. Then the unipotent characters of L I are {Id, St, ρ θ , ρ θε , ρ r , θ 10 } where θ 10 is the unique unipotent cuspidal character. Using [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF]Theorem 4.3.4] we obtain

H U I \X W I w 0 = (h 2 t + h) 2 h 4 St + h 5 t 2 (ρ θ + ρ θε + 2θ 10 ) + h 8 t 4 Id = h 6 St + h 7 2tSt + t 2 (ρ θ + ρ θε + 2θ 10 ) + h 8 t 2 St + 2t 3 (ρ θ + ρ θε + 2θ 10 ) + h 9 t 4 (ρ θ + ρ θε + 2θ 10 ) + h 10 t 4 Id + 2h 11 t 5 Id + h 12 t 6 Id
and also

H U I \X W I w 0 t 1 t 2 = h 2 t(h 2 t + h) 2 h 2 (St + tθ 10 ) + h 3 tρ r + h 4 t 2 Id = h 6 (tSt + t 2 θ 10 ) + h 7 t 2 (2St + ρ r ) + 2t 3 θ 10 + h 8 t 3 (St + 2ρ r + Id) + t 4 θ 10 + h 9 t 4 (ρ r + 2Id) + h 10 t 5 Id.
We observe that the unipotent characters ρ θ , ρ θε and ρ r appear in the cohomology of only one of the two varieties. Using the long exact sequence associated to the decomposition

U I \X(w) = U I \X W I w 0 ∪ U I \X W I w 0 t 1 t 2 ∪ U I \X W I w 0 t 4 t 3
we deduce the isotypic part of these characters in the cohomology of U I \X(w). It is given by

h 7 t 2 (ρ θ + ρ θε + 2ρ r ) + h 8 t 3 (2ρ θ + 2ρ θε + 4ρ r ) + h 9 t 4 (ρ θ + ρ θε + 2ρ r ).
(2.1)

The isotypic parts for the unipotent characters St and Id fit into the following exact sequences:

0 -→ St -→ H 6 c U I \X(w) St -→ 2tSt -→ 2tSt -→ H 7 c U I \X(w) St -→ 4t 2 St -→ t 2 St -→ H 8 c U I \X(w) St -→ 2t 3 St -→ 0 0 -→ H 8 c U I \X(w) Id -→ 2t 3 Id -→ 0 0 -→ H 9 c U I \X(w) Id -→ 4t 4 Id -→ t 4 Id -→ H 10 c U I \X(w) Id -→ 2t 5 Id -→ 2t 5 Id -→ H 11 c U I \X(w) Id -→ 0 0 -→ t 6 Id -→ H 12 c U I \X(w) Id -→ 0
h 6 St + h 7 t 2 (3St + ρ θ + ρ θε + 2ρ r ) + h 8 t 3 (2St + 2ρ θ + 2ρ θε + 4ρ r + 2Id) + h 9 t 4 (ρ θ + ρ θε + 2ρ r + 3Id) + h 12 t 6 Id.
Remark 2.3. The long exact sequence coming from the decomposition of the variety U I \X(w) does not give enough information to determine the θ 10 -isotypic part:

0 -→ H 6 c U I \X(w) θ 10 -→ 2t 2 θ 10 -→ 2t 2 θ 10 -→ H 7 c U I \X(w) θ 10 -→ 4t 3 θ 10 -→ 4t 3 θ 10 -→ H 8 c U I \X(w) θ 10 -→ 2t 4 θ 10 -→ 2t 4 θ 10 -→ H 9 c U I \X(w) θ 10 -→ 0.
One could nonetheless hope that in this particular situation the boundary maps are isomorphisms, which would imply that θ 10 cannot appear in the cohomology of U I \X(w). This will be the case if and only if the graded endomorphism ring End G (H • c (X(w), Q ℓ )) is concentrated in degree 0.

Cuspidal characters.

From [START_REF] Broué | Generic blocks of finite reductive groups[END_REF] we know that the irreducible constituents of the alternating sum of the comology of X(w) are the unipotent characters in the principal Φ 8 -block, namely {Id G , St G , φ 9,10 , φ 16,5 , φ 9,2 } for the principal series and {F 4 [-1], F 4 [i], F 4 [-i]} for the cuspidal characters (with the notation in [START_REF] Carter | Finite groups of Lie type[END_REF]). We observe that the restriction of these characters to L I are exactly the one obtained in the previous proposition. Since the Harish-Chandra restriction preserves the Harish-Chandra series, we can deduce the contribution of the principal series to the cohomology of X(w). The missing ones are either in the series associated to θ 10 -which we could not determineor are cuspidal characters. We shall deduce the contribution of the latter using the results in Section 1.2.

Recall that G has 7 cuspidal unipotent characters, namely

F 4 [-1], F 4 [i], F 4 [-i], F 4 [θ], F 4 [θ 2 ], F I
4 [START_REF] Bessis | Finite complex reflection arrangements are k(π, 1)[END_REF] and F II 4 [START_REF] Bessis | Finite complex reflection arrangements are k(π, 1)[END_REF] where i (resp. θ) is a primitive 4th root of unity (resp. a primitive 3rd root of unity). Let ρ be a cuspidal unipotent character and let v ≤ w. By cuspidality ρ cannot appear in the cohomology of Deligne-Lusztig varieties associated to elements lying in a proper parabolic subgroup of W. In particular it cannot appear in the cohomology of X(v) or X(v) unless v is in the following set

V = w, t 1 t 2 t 3 t 2 t 4 , t 1 t 3 t 2 t 3 t 4 , t 1 t 2 t 3 t 4 , t 1 t 3 t 2 t 4 . Define Z = X(t 1 t 2 t 3 t 2 t 4 ) ∪ X(t 1 t 3 t 2 t 3 t 4 ) and Z ′ = X(t 1 t 2 t 3 t 4 ) ∪ X(t 1 t 3 t 2 t 4 )
. The property (C2) yields the following exact sequences:

• • • -→ H i c X(w) ρ -→ H i c X(w) ρ -→ H i c Z ρ -→ • • • (2.4) • • • -→ H i c X(t 1 t 2 t 3 t 2 t 4 ) ρ -→ H i c X(t 1 t 2 t 3 t 2 t 4 ) ρ -→ H i c Z ′ ρ -→ • • • (2.5) • • • -→ H i c X(t 1 t 3 t 2 t 3 t 4 ) ρ -→ H i c X(t 1 t 3 t 2 t 3 t 4 ) ρ -→ H i c Z ′ ρ -→ • • • (2.6)
Moreover, one can check that each of these compactifications is actually rationally smooth, and therefore one can use (C3) to compute the cuspidal part of their cohomology, denoted by H X (t 1/2 , h). They are given by

H X(w) = h 6 t 3 F 4 [-1] + F 4 [i] + F 4 [-i] + 2F 4 [θ] + 2F 4 [θ 2 ] (2.7) and H X(t 1 t 2 t 3 t 2 t 4 ) = H X(t 1 t 3 t 2 t 3 t 4 ) = (h 4 t 2 + h 6 t 3 ) F 4 [i] + F 4 [-i] + F 4 [θ] + F 4 [θ 2 ] .
Furthermore, the elements t 1 t 2 t 3 t 4 and t 1 t 3 t 2 t 4 are minimal in the set V for the Bruhat order, so that for any unipotent cuspidal character ρ

H i c (Z ′ ) ρ ≃ H i c X(t 1 t 2 t 3 t 4 ) ρ ⊕ H i c X(t 1 t 3 t 2 t 4 ) ρ ≃ H i c X(c) ⊕2 ρ
where c is any Coxeter element of W. Using [20, table 7.3] we deduce that

H Z ′ = 2h 4 t 2 F 4 [i] + F 4 [-i] + F 4 [θ] + F 4 [θ 2 ] .
Together with 2.5 and 2.6, and the fact that the cohomology of X(t 1 t 2 t 3 t 2 t 4 ) and X(t 1 t 3 t 2 t 3 t 4 ) vanishes in degree 4, we obtain

H X(t 1 t 2 t 3 t 2 t 4 ) = H X(t 1 t 3 t 2 t 3 t 4 ) = (h 5 t 2 + h 6 t 3 ) F 4 [i] + F 4 [-i] + F 4 [θ] + F 4 [θ 2 ] .
From these results, one can now partially determine the cohomology of Z: for any unipotent cuspidal character, we use the following exact sequence

• • • -→ H i c X(t 1 t 2 t 3 t 2 t 4 ) ρ ⊕ H i c X(t 1 t 3 t 2 t 3 t 4 ) ρ -→ H i c Z ρ -→ H i c Z ′ ρ -→ • • • to deduce that there exist integers 0 ≤ ε i ≤ 2 such that H Z = (h 4 + h 5 )t 2 ε 1 F 4 [i] + ε 2 F 4 [-i] + ε 3 F 4 [θ] + ε 4 F 4 [θ 2 ] + 2h 6 t 3 F 4 [i] + F 4 [-i] + F 4 [θ] + F 4 [θ 2 ] .
However 2.4 forces each character ε i ρ to be a component of H 5 c X(w) since H 4 c X(w) is zero by 2.7. But the cohomology of X(w) vanishes outside the degrees 6, . . ., 12, and hence the ε i 's must be zero. Consequently, the exact sequence 2.4 can be decomposed into

0 -→ H 6 c X(w) F 4 [-1] -→ t 3 F 4 [-1] -→ 0 0 -→ H 6 c X(w) F 4 [±i] -→ t 3 F 4 [±i] -→ 2t 3 F 4 [±i] -→ H 7 c X(w) F 4 [±i] -→ 0 0 -→ H 6 c X(w) F 4 [θ j ] -→ 2t 3 F 4 [θ j ] -→ 2t 3 F 4 [θ j ] -→ H 7 c X(w) F 4 [θ j ] -→ 0.
We use (C4) to conclude: the characters F 4 [±i], and F 4 [θ j ] already occur in the cohomology of the Deligne-Lusztig variety associated to a Coxeter element. Since w is not F-conjugate to a Coxeter element, they cannot appear in H 6 c X(w) with an eigenvalue of absolute value q 3 , and the previous exact sequences determine the cuspidal part of the cohomology of X(w). Proposition 2.8. Let w = t 1 t 2 t 3 t 2 t 4 . The cuspidal part of the cohomology of X(w) is given by bH i X(w), Q ℓ St q 2 φ 9,10 q 3 φ 16,5 q 4 φ 9,2

h 6 t 3 F 4 [-1] + h 7 t 3 F 4 [i] + F 4 [-i]).

Cohomology of X(w). The unipotent characters in the principal

q 6 Id -q 3 F 4 [-1] iq 3 F 4 [i] -iq 3 F 4 [-i]
2.2 9-regular elements for groups of type E 6

In this section we assume that (G, F) is a split group of type E 6 . The largest regular number (excluding the Coxeter number) being 9, we are interested in computing the cohomology of X(w) for any 9th root of π, or equivalently for any good 9-regular element. We will label the simple reflections as follows

t 1 t 3 t 2 t 4 t 5 t 6
As before, we may, and we will, consider a particular root of π, since the cohomology of the corresponding Deligne-Lusztig variety X(w) does not depend on this choice. We set w = t 1 t 3 t 4 t 3 t 2 t 4 t 5 t 6 .

Cohomology of U I \X(w)

. We decompose the quotient of X(w) by U I for I = {t 2 , t 3 , t 4 , t 5 }. The situation is similar to the one studied in Section 2.1.1: a piece X x is non-empty if and only if W I x is one of the three cosets among W I w 0 , W I w 0 t 6 t 5 t 4 and W I w 0 t 1 t 3 .

• Let J = S {t 1 }. We have W J w 0 t 1 t 3 = W J w 0 and therefore the piece corresponding to W I w 0 t 6 t 5 t 4 is stable by the action of P J . Let y be the minimal element of W J w 0 t 6 t 5 t 4 . Since w 0 (t 6 t 5 t 4 ) = t 1 t 3 t 4 is J-reduced, y = w J w 0 t 6 t 5 t 4 . Let us decompose w as w = w 1 w 2 w 3 with w 1 = t 1 , w 2 = t 3 = t 3 w ′ 2 and w 3 = t 4 t 3 t 2 t 4 t 5 t 6 . Then y w 1 = t 6 , y w ′ 2 = 1 and y w 3 = t 3 t 5 t 4 t 2 are all elements of W J . In addition, they satisfy the assumptions of Theorem 1.1 (see also Remark 1.2) so that

H • c (X W J y , Q ℓ ) U J ≃ H • c (G a × G m × X L J (t 6 t 3 t 5 t 4 t 2 ), Q ℓ ).
Now X L J (t 6 t 3 t 5 t 4 t 2 ) is a Deligne-Lusztig variety associated to a Coxeter element, and therefore the cohomology of its quotient by U I ∩ L J is given by [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF]. We obtain

H • c (X W I w 0 t 6 t 5 t 4 , Q ℓ ) U I ≃ H • c (G a × G m × X L J (t 6 t 3 t 5 t 4 t 2 ), Q ℓ ) U I ∩L J ≃ H • c (G a × (G m ) 2 × X L I (t 2 t 5 t 4 t 3 ), Q ℓ ).
• For the piece X W I w 0 t 1 t 3 , we proceed as above: let K = S {t 6 } and z be the minimal element of W K w 0 t 1 t 3 . It is clearly z = w K w 0 t 1 t 3 since t 6 t 5 is K -reduced. We can decompose w as w = w 1 w 2 where w 1 = t 1 t 3 t 4 t 3 t 2 and w 2 = t 4 (t 5 t 6 ) = t 4 w ′ 2 . We have z w 1 = t 2 t 4 t 5 and z w ′ 2 = t 3 t 1 so that with Theorem 1.1 and [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] we obtain

H • c (X W I w 0 t 1 t 3 , Q ℓ ) U I ≃ H • c (G a × G m × X L K (t 2 t 4 t 5 t 3 t 1 ), Q ℓ ) U I ∩L K ≃ H • c (G a × (G m ) 2 × X L I (t 5 t 4 t 2 t 3 ), Q ℓ ).
• For the open piece X W I w 0 we can directly apply Theorem 1.1 by decomposing w as w = w 1 w 2 with w 1 = t 1 (t 3 t 4 t 3 t 2 t 4 t 5 ) and w 2 = t 6 . We deduce

H • c (X W I w 0 , Q ℓ ) U I ≃ H • c ((G m ) 2 × X L I (t 5 t 4 t 5 t 2 t 4 t 3 ), Q ℓ ).
By the properties of the Bruhat order the varieties X W I w 0 t 6 t 5 t 4 and X W I w 0 t 1 t 3 are both closed subvarieties of X(w). Therefore the cohomology of the union X f = U I \X W I w 0 t 6 t 5 t 4 ∪ U I \X W I w 0 t 1 t 3 can be deduced from [20, Table 7.3] whereas the cohomology of X o = U I \X W I w 0 is given by [9, Theorem 12.4]:

H X o = (h 2 t + h) 2 h 6 St + h 7 t 2 (ρ 1 2 + + ρ 1 2 -+ ρ 21 2 ) + 2t 3 D 4 + 2h 8 t 3 ρ 1.21 + h 9 t 4 (ρ 2+ + ρ 2-+ ρ 31 ) + h 12 t 6 Id H X f = 2h 2 t(h 2 t + h) 2 h 4 (St + t 2 D 4 ) + h 5 tρ 1.1 3 + h 6 t 2 ρ 1 2 .2 + h 7 t 3 ρ 1.3 + h 8 t 4 Id
where ρ λ is the unipotent character (in the principal series) associated to the character λ of W I and D 4 is the unique unipotent cuspidal character of L I .

As before, any character in the principal series which is different from St and Id cannot appear in the cohomology of both of the varieties, so that the isotypic part on the cohomology of U I \X(w) is the sum of the isotypic part on H

Cuspidal characters.

The group G has only two cuspidal characters, denoted by E 6 [θ] and E 6 [θ 2 ] where θ is a primitive 3rd root of unity. In order to determine they contribution to the cohomology of X(w) , we use the compactifications X(v) for v ≤ w. However, unlike the type F 4 , they are not always rationally smooth and we shall rather work with "bigger" compactifications, obtained by underlining all the simple reflections. For details on the explicit computations we refer to Section 2.1.2. We start by defining the following closed subvariety of X(w): Z = X(t 1 t 4 t 3 t 2 t 4 t 5 t 6 ) ∪ X(t 1 t 3 t 3 t 2 t 4 t 5 t 6 ) ∪ X(t 1 t 3 t 4 t 2 t 4 t 5 t 6 ) ∪ X(t 1 t 3 t 4 t 3 t 2 t 5 t 6 ) so that we obtain, for any cuspidal character ρ, a long exact sequence

• • • -→ H i c X(w) ρ -→ H i c X(w) ρ -→ H i c Z ρ -→ • • • (2.12)
We determine the cuspidal part of Z as follows: we compute, for any element v

X(v) = H X(v) = (h 7 t 3 + h 8 t 4 ) E 6 [θ] + E 6 [θ 2 ]
by means of the following exact sequences

• • • -→ H i c X(v) ρ -→ H i c X(v) ρ -→ H i c (X(c)) ρ ⊕2 -→ • • •
and the precise values and

H X(v) = (h 6 t 3 + h 8 t 4 ) E 6 [θ] + E 6 [θ 2 ] H X(c) = h 6 t 3 E 6 [θ] + E 6 [θ 2 ]
that can be found using (C3). Note that we have also used the fact that the cohomology of X(v) is zero outside the degrees 7, . . ., 14. For the element v = t 1 t 3 t 3 t 2 t 4 t 5 t 6 we use [11, Proposition 3.2.10] and we obtain the same value again:

H X(v) = (h 2 t + h) H X(c) = (h 7 t 3 + h 8 t 4 ) E 6 [θ] + E 6 [θ 2 ]
In particular, the cohomology of Z fits into the following long exact sequence

• • • -→ H i c (X(v)) ρ ⊕4 -→ H i c (Z) ρ -→ H i c (X(c)) ρ ⊕4 -→ • • • We claim that H Z = 4h 8 t 4 E 6 [θ] + E 6 [θ 2 ] . (2.13)
Again, the exact sequence itself is not enough to compute this value, but it can be deduced from the following properties:

• the cohomology of X(w) vanishes in degree 7 by (C1);

• H X(w) = 3h 8 t 4 E 6 [θ] + E 6 [θ 2 ] which forces in particular H 6 c (X(w)) to have no cuspidal constituent. These properties, together with 2.12, ensure that the coefficient of h 6 in H Z is zero, and we deduce 2.13.

Consequently, the decomposition X(w) = X(w) ∪ Z yields the following exact sequence for any cuspidal character ρ:

0 -→ H 8 c (X(w)) ρ -→ 3t 4 ρ -→ 4t 4 ρ -→ H 9 c (X(w)) ρ -→ 0.
Finally, by (C4) the group H 8 c (X(w)) cannot contain any unipotent cuspidal character with an eigenvalue of absolute value q 4 and we obtain: Proposition 2.14. Let w = t 1 t 3 t 4 t 3 t 2 t 4 t 5 t 6 . The contribution of the cuspidal characters of G to the cohomology of X(w) is given by

h 9 t 4 E 6 [θ] + E 6 [θ 2 ] .

Cohomology of X(w)

. By [START_REF] Broué | Blocs et séries de Lusztig dans un groupe réductif fini[END_REF], the irreductible constituents of the virtual character associated to the cohomology of X(w) are exactly the unipotent characters in the principal Φ 9 -block, namely b uni = Id G , St G , φ 20,20 , φ 64,13 , φ 90,8 , φ 64,4 ,

φ 20,2 , E 6 [θ], E 6 [θ 2 ]
. By looking at the Harish-Chandra restriction of these characters, we can deduce from Proposition 2.10 and 2.14 the following theorem: Theorem 2.15. Let (G, F) be a split group of type E 6 and w be a good 9-regular element of W. The contribution to the cohomology of the Deligne-Lusztig X(w) of the principal series and the cuspidal characters coincides with the contribution of the principal Φ 9 -block, and it is given by i 8 9 10 11 12 13 14 15 16

bH i (X(w), Q ℓ ) St q 2 φ 20,20 q 3 φ 64,13 q 4 φ 90,8 q 5 φ 64,4 q 6 φ 20,2 q 8 Id

θq 4 E 6 [θ] θ 2 q 4 E 6 [θ 2 ]
Conjecturally, for good regular elements, there should be no cancelation in the virtual character (-

1) i H i c (X(w), Q ℓ ) ∈ K 0 (G-mod) [3, Conjecture 5.7].
In particular, the series associated to the cuspidal character of D 4 should not appear in the cohomology of X(w): Assumption 2.16. For good 9-regular elements in E 6 , the cohomology of X(w) has no constituent in the Harish-Chandra series associated to the cuspidal representation of D 4 .

This assumption will be essential to study the contribution of the D 4 -series for groups of type E 7 and E 8 (see Theorem 2.20 and 2.26).

14-regular elements for groups of type E 7

We now assume that (G, F) is a split group of type E 7 and we are interested in computing the cohomology of Deligne-Lusztig varieties associated to good 14regular elements. We will label the simple reflections according to the following Dynkin diagram

t 1 t 3 t 2 t 4 t 5 t 6 t 7
and consider a specific 9th root of π: w = t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 .

Cohomology of U I \X(w)

. Let I = S {t 7 }. The group L I has type E 6 and we can use the results in the previous section to compute the cohomology of the quotient of X(w) by U I . In the decomposition of X(w) by P I -cosets in G/B, only two pieces are non-empty, with associated cosets W I w 0 and W I w 0 t 7 t 6 t 5 . We can apply Theorem 1.1 in these two cases:

• when y = w I w 0 t 7 t 6 t 5 we decompose w as w = w 1 w 2 with w 1 = t 7 t 6 t 5 t 4 t 5 t 2 and w 2 = t 4 (t 3 t 1 ) = t 4 w ′ 2 . We have y w 1 = t 1 t 3 t 4 t 2 and y w ′ 2 = t 5 t 6 so that

H • c (X W I y , Q ℓ ) U I ≃ H • c (G a × G m × X L I (t 1 t 3 t 4 t 2 t 5 t 6 ), Q ℓ ).
• for x = w I w 0 we observe that w = t 7 (t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 ) = t 7 w ′ with x w ′ ∈ W J and deduce that

H • c (X W I w 0 , Q ℓ ) U I ≃ H • c (G m × X L I (t 1 t 3 t 4 t 3 t 2 t 4 t 5 t 6 ), Q ℓ ).
The cohomology of these varieties is known by Theorem 2.15 and [20, Table 7.3].

Recall that for any Coxeter element c I of W I , the cohomology of the corresponding variety is given by The case of the Harish-Chandra series associated to the cuspidal character of D 4 remains undetermined unless we know the contribution of this series to the cohomology of the open part. However, in our situtation, none of these characters should appear, and the isotypic part on the cohomology of the union U I \X(w) should come from the Coxeter variety only. Proposition 2.18. Assume that 2.16 holds, and let w = t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 and I = ∆ {t 7 }. Then the contribution of the Harish-Chandra series associated to the cuspidal character of D 4 to the cohomology of U I \X(w) is given by

H X L I (c I ) = h 6 St + t 2 D 4,ε + t 3 E 6 [θ] + t 3 E
h 9 t 3 D 4,ε + h 10 t 4 D 4,ε + D 4,r + h 11 t 5 D 4,r + D 4,Id + h 12 t 6 D 4,Id .
Finally, for the cuspidal characters E 6 [θ] and E 6 [θ 2 ], we have a long exact sequence

0 -→ H 9 c U I \X(w) E 6 [θ] -→ t 4 E 6 [θ] -→ t 4 E 6 [θ] -→ H 10 c U I \X(w) E 6 [θ] -→ t 5 E 6 [θ] -→ t 5 E 6 [θ] -→ H 11 c U I \X(w) E 6 [θ] -→ 0.
This is not enough to determine their contribution and we can only hope that they actually do not appear in the cohomology of U I \X(w).

Cuspidal characters.

The group G has only two cuspidal unipotent characters, namely E 7 [i] and E 7 [-i], where i is a primitive 4th root of unity. The method to determine their contribution to the cohomology is strictly identical to the case of E 6 and yields bH i (X(w), Q ℓ ) St q 2 φ 27,37 q 3 φ 105,26 q 4 φ 189,17 q 5 φ 189,10

-q 3 D 4,1 3 . -q 4 D 4,1 2 .1 -q 5 D 4,1.2 -q 6 D 4,.3 iq 9/2 E 7 [i] -iq 9/2 E 7 [-i] i 14 15 16 17 18 bH i (X(w), Q ℓ ) q 6 φ 105,5 q 7 φ 27,2 q 9 Id
where the D 4 -series is given under Assumption 2.16.

In our situation, the non-cancellation for the corresponding Deligne-Lusztig virtual character is equivalent to the following: Assumption 2.21. The characters lying in the Harish-Chandra series associated to the cuspidal characters E 6 [θ] and E 6 [θ 2 ] do not appear in the cohomology of X(w).

24-regular elements for groups of type E 8

We close this section by studying the cohomology of Deligne-Lusztig varieties associated to good 24-regular elements in E 8 . We will label the simple reflections as follows

t

Cohomology of U I \X(w).

The situation is very similar to the case of E 7 so we will omit the details. When I = S {t 8 }, the pieces corresponding to W I w 0 and W I w 0 t 8 t 7 t 6 t 5 are the only non-empty pieces, and the cohomology of their quotient by U I is given by ], E I 8 [START_REF] Bessis | Finite complex reflection arrangements are k(π, 1)[END_REF], E II 8 [START_REF] Bessis | Finite complex reflection arrangements are k(π, 1)[END_REF] and E 8 [ζ j ] where ζ is a primitive 5th root of unity and j = 1, 2, 3, 4. We proceed as in the previous cases to determine they contribution to the cohomology of X(w). However, due to the large number of cuspidal characters, the calculations are a bit more tedious.

H • c (U I \X W I w 0 t 8 t 7 t 6 t 5 , Q ℓ ) ≃ H • c G a × G m × X L I (t 7 t 6 t 5 t 4 t 2 t 3 t 1 ), Q ℓ and H • c (U I \X W I w 0 , Q ℓ ) ≃ H • c G m × X L I (
We start by considering the closed subvariety Z of X(w) consiting of the union of the varieties X(v) where v runs over the set The cohomology of this variety fits in the following long exact sequence, for any cuspidal character ρ

t
• • • -→ H i c X(w) ρ -→ H i c X(w) ρ -→ H i c Z ρ -→ • • • (2.24)
The elements of the Braid monoid obtained by un-underlying the elements v will be denoted by v 1 , v 2 , v 3 et v 4 . Note that only v 4 is not the canonical lift of an element of W. For j = 1, 2, 3, the cuspidal part of the cohomology of X(v j ) ≃ X(v j ) can be deduced from the following exact sequence

• • • -→ H i c X(v j ) ρ -→ H i c X(v j ) ρ -→ H i c (X(c)) ρ ⊕2 -→ • • •
together with the following properties

• the cuspidal part of H • c (X(v j )) can be explicitely computed using (C3):

H X(v j ) = (h 8 t 4 + h 10 t 5 ) E 8 [-θ] + E 8 [-θ 2 ] + E 8 [ζ] + E 8 [ζ 2 ] + E 8 [ζ 3 ] + E 8 [ζ 4 ]
• the cuspidal part of a variety associated to a Coxeter element is given by [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] (or equivalently can be computed using (C3)):

H X(c) = h 8 t 4 E 8 [-θ] + E 8 [-θ 2 ] + E 8 [ζ] + E 8 [ζ 2 ] + E 8 [ζ 3 ] + E 8 [ζ 4 ]
• the cohomology of X(v j ) vanishes in degree 8.

We obtain, for j = 1, 2, 3:

H X(v j ) = (h 9 t 4 + h 10 t 5 ) E 8 [-θ] + E 8 [-θ 2 ] + E 8 [ζ] + E 8 [ζ 2 ] + E 8 [ζ 3 ] + E 8 [ζ 4 ] .
Using [11, Proposition 3.2.10], one can check that it is also the cuspidal part of the cohomology of X(v 4 ).

We claim that we can derive the cohomology of Z: for any cuspidal character ρ, we have an exact sequence

• • • -→ 4 j=1 H i c X(v j ) ρ -→ H i c Z ρ -→ H i c (X(c)) ρ ⊕4 -→ • • •
Furthermore, the cohomology of X(w) vanishes in degree 9 and the cuspidal part of H • c (X(w)) is concentrated in degree 10, given by

h 10 t 5 E 8 [i] + E 8 [-i] + 3(E 8 [-θ] + E 8 [-θ 2 ]) + 4(E 8 [ζ] + E 8 [ζ 2 ] + E 8 [ζ 3 ] + E 8 [ζ 4 ])
Consequently, the cuspidal part of H 8 c (Z) is zero by 2.24 and we obtain

H Z = 4h 10 t 5 E 8 [-θ] + E 8 [-θ 2 ] + E 8 [ζ] + E 8 [ζ 2 ] + E 8 [ζ 3 ] + E 8 [ζ 4 ] .
In particular, we can unpack the exact sequence 2.24 according to the different cuspidal characters as follows

0 -→ H 10 c X(w) E 8 [±i] -→ t 5 E 8 [±i] -→ 0 0 -→ H 10 c X(w) E 8 [-θ i ] -→ 3t 5 E 8 [-θ i ] -→ 4t 5 E 8 [-θ i ] -→ H 11 c X(w) E 8 [-θ i ] -→ 0 0 -→ H 10 c X(w) E 8 [ζ j ] -→ 4t 5 E 8 [ζ j ] -→ 4t 5 E 8 [ζ j ] -→ H 11 c X(w) E 8 [ζ j ] -→ 0
To conclude, we observe that the unipotent characters E 8 [-θ i ] and E 8 [ζ j ] already appear in the Coxeter variety, and for that reason they cannot be constituents of H 10 c (X(w)) with an eigenvalue of absolute value q 5 (see (C4)). ,

E 6 [θ] φ ′ 1,3 , E 6 [θ] φ 2,2 , E 6 [θ] φ ′′ 1,3 , E 6 [θ 2 ] φ ′ 1,3 , E 6 [θ 2 ] φ 2,2 , E 6 [θ 2 ] φ ′′ 1,3 , E 8 [i], E 8 [-i], E 8 [-θ], E 8 [-θ 2 ]
By comparing the restriction to E 7 of these characters and Proposition 2.22, 2.23 and 2.25 we obtain a good approximation of the cohomology of X(w).

Theorem 2.26. Let (G, F) be a split group of type E 8 and w be a good 24-regular element of W. The contribution to the cohomology of the Deligne-Lusztig X(w) of the principal series, the D 4 -series, the E 6 -series and the cuspidal characters coincides with the contribution of the principal Φ 24 -block, and it is given by i 10 11 12 13

bH i (X(w), Q ℓ ) St q 2 φ 35,74 q 3 φ 160,55 q 4 φ 350,38

-q 3 D 4,φ ′′ 2,16 -q 4 D 4,φ ′′ 8,9 -q 5 D 4,φ 12,4 -q 6 D 4,φ ′ 8,3 θq 4 E 6 [θ] φ ′ 1,3 θq 5 E 6 [θ] φ 2,2 θq 6 E 6 [θ] φ ′′ 1,3 θ 2 q 4 E 6 [θ 2 ] φ ′ 1,3 θ 2 q 5 E 6 [θ 2 ] φ 2,2 θ 2 q 6 E 6 [θ 2 ] φ ′′ 1,3 iq 5 E 8 [i] -iq 5 E 8 [-i] -θq 5 E 8 [-θ] -θ 2 q 5 E 8 [-θ 2 ] i 14 15 16 17 18 19 20
bH i (X(w), Q ℓ ) q 5 φ 448,25 q 6 φ 350,14 q 7 φ 160,7 q 8 φ 35,2

q 10 Id -q 7 D 4,φ ′ 2,4
where the D 4 -series is given under Assumption 2. [START_REF] Geck | Brauer trees of Hecke algebras[END_REF] and the E 6 -series under Assumption 2.21.

Conjectures on associated Brauer trees

Having computed the cohomology of some Deligne-Lusztig varieties for exceptionals groups, we would like to propose conjectures on Brauer trees for the corresponding principal Φ d -blocks.

Recall from [START_REF] Broué | Generic blocks of finite reductive groups[END_REF] that if d is a regular number, and w is a d-regular element, the irreducible constituent of the virtual character R

G T w (1) = (-1) i H c (X(w), Q ℓ ) are exactly the unipotent characters in the principal Φ d -block. If moreover C W (wF) ≃ N G (T w )/C G (T w ) is cyclic, then the Φ d -block is generically of cyclic defect: if ℓ divides Φ d (q) but
does not divide |W|, then any Sylow subgroup of G is cyclic. In that case, the representation theory of the block (i.e. the module category over the block) can be decribed by its Brauer tree. More precisely, in this situation:

• any ℓ-character θ of T w is in general position and the associated irreducible character χ θ = (-1) ℓ(w) R G T w (θ) is cuspidal by [START_REF] Lusztig | Representations of finite Chevalley groups[END_REF]Proposition 2.18]. Moreover, using [START_REF] Digne | Representations of finite groups of Lie type[END_REF]Proposition 12.2] it can be shown that its restriction to the set of ℓ-regular elements is independent from θ. Any character of this form is said to be a exceptional;

• there are e = |C W (w)| unipotent characters {χ 0 , . . ., χ e-1 } in the block, which will be refered as the non-exceptional characters;

Now if we consider the sum χ exc of all distinct unipotent characters, any projective indecomposable F ℓ G-module lifts uniquely, up to isomorphism, to a Z ℓmodule P whose character is [P] = χ + χ ′ for χ, χ ′ two distinct characters in V = {χ exc , χ 0 , . . ., χ e-1 }. We define the Brauer tree Γ of the block to be the graph with vertices labelled by V and egdes χχ ′ whenever there exists a projective indecomposable module with character χ+χ ′ . This graph is a tree and its planar embbeding determines the module category over the block up to Morita equivalence.

When d = h is a the Coxeter number, Hiss, Lübeck and Malle have formulated in [START_REF] Hiss | The Brauer trees of the exceptional Chevalley groups of type E 6[END_REF] a conjecture relating the cohomology of the Deligne-Lusztig variety associated to a Coxeter element (together with the action of F) and the planar embedded Brauer tree of the principal Φ h -block. Using the explicit results on the cohomology of Deligne-Lusztig varieties that we have obtained, and the Brauer trees that we already already know from [START_REF] Hiss | The Brauer trees of the exceptional Chevalley groups of types F 4 and 2 E 6[END_REF] and [START_REF] Hiss | The Brauer trees of the exceptional Chevalley groups of type E 6[END_REF], we shall propose two conjectural Brauer trees for groups of type E 7 and E 8 .

Observations

Let (G, F) be a split group of type F 4 and w be a good 8-regular element. When ℓ divides Φ 8 (q) and does not divide the order of W, we can observe that the classes in F ℓ of the eigenvalues of F on bH • c (X(w), Q ℓ ) form the group of 8th roots of unity, generated by the class of q. Therefore to any non-exceptional character χ one can associate an integer j χ such that the class of the corresponding eigenvalue of F coincides with the class of q j χ . By [START_REF] Hiss | The Brauer trees of the exceptional Chevalley groups of types F 4 and 2 E 6[END_REF], the Brauer tree of the block, together with the integers j χ is given by Now assume that (G, F) is a split group of type E 6 . The Brauer tree of the principal Φ 9 -block of G has been determined in [START_REF] Hiss | The Brauer trees of the exceptional Chevalley groups of type E 6[END_REF]. It corresponds to the following picture: Remark 3.1. Unlike the Coxeter case (see [START_REF] Dudas | Coxeter Orbits and Brauer trees I[END_REF] and [START_REF] Dudas | Coxeter Orbits and Brauer trees II[END_REF]), the cohomology of the Deligne-Lusztig variety X(w) with coefficients in Z ℓ is not torsion-free. Indeed, it is impossible to represent the generalized (q 2 )-eigenspace of F on X(w) with a complex of projective modules 0 -→ P f -→ Q -→ 0 where the cokernel of f is torsion-free. Note that even the cohomology of the complex constructed by Rickard in [START_REF] Rickard | Derived categories and stable equivalence[END_REF]Section 4] will also have a non-trivial torsion part (one can show nevertheless that the torsion is always cuspidal).

Conjectures

From the results obtained in Theorem 2.20 and 2.26, it is not difficult to extrapolate the previous trees to the case of E 7 and E 8 . We conjecture that the Brauer trees of the principal Φ 14 -block in E 7 and the principal Φ 24 -block in E 8 are given by Figure 3 and4. Note that • the lines represented by each Harish-Chandra series, as well as the real steam, are known from [START_REF] Geck | Brauer trees of Hecke algebras[END_REF];

• the simple modules corresponding to edges connecting two different series are necessarily cuspidal.

Theorem 2 . 9 .

 29 Φ 8block b are given by b uni = Id, St G , φ 9,10 , φ 16,5 , φ 9,2 , F 4 [-1], F 4 [i], F 4 [-i] . From Proposition 2.2 and 2.8 we deduce the contribution to the cohomology of X(w) of any unipotent character in the block: Let (G, F) be a split group of type F 4 and w be a good 8-regular element. The contribution to the cohomology of the Deligne-Lusztig X(w) of the principal series and the cuspidal characters coincides with the contribution of the principal Φ 8 -block, and it is given by i

F 4 1 F 4 [-i] 5 Figure 1 :

 41451 Figure 1: Brauer tree of the principal Φ 8 -block of F 4

7 Figure 2 :

 72 Figure 2: Brauer tree of the principal Φ 9 -block of E 6

Figure 3 :

 3 Figure 3: Brauer tree of the principal Φ 14 -block of E 7

Proposition 2.2. Let

  Any morphism above is F-equivariant so that we can consider each power of t separately. On the other hand, the only unipotent character of G whose restriction is St L I (resp.Id L I ) is St G (resp. Id G ). But from[START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF] Proposition 3.3.14 and 3.3.15] we know exactly where these characters can appear in the cohomology of X(w) as well as the corresponding eigenvalue of F. Using 2.1 we deduce that tSt (resp. t 2 St) cannot appear in H6 With the previous exact sequences, this forces the isotypic part of St and Id in the cohomology of U I \X(w) to be h6 St + 3h 7 t 2 St + h 8 t 3 (2St + 2Id) + 3h 9 t 4 Id + h 12 t 6 Id. w = t 1 t 2 t 3 t 2 t 3 t 4 and I = {t 2 , t 3 }. The characters of the principal series in the cohomology of U I \X(w) are given by

	Together with 2.1 we finally obtain

c U I \X(w) or in H 7 c U I \X(w) (resp. in H 8 c U I \X(w) ) and that t 4 Id (resp. t 5 Id) cannot appear in H 10 c U I \X(w) (resp. in H 11 c U I \X(w) ).

  Let w = t 1 t 3 t 4 t 3 t 2 t 4 t 5 t 6 and I = {t 2 , t 3 , t 4 , t 5 }. The contribution of the characters in the principal series to the cohomology of U I \X(w) is given by h 8 St + h 9 t 2 3St + ρ 1 2 + + ρ 1 2 -+ ρ 21 2 + 2ρ 1.1 3 + h 10 t 3 2St + 2ρ 1 2 + + 2ρ 1 2 -+ 2ρ 21 2 + 4ρ 1.1 3 + 2ρ 1.21 + 2ρ 1 2 .2 + h 11 t 4 ρ 1 2 + + ρ 1 2 -+ ρ 21 2 + 2ρ 1.1 3 + 4ρ 1.21 + 4ρ 1 2 .2 + ρ 2+ + ρ 2-+ ρ 31 + 2ρ 1.3 + h 12 t 5 2ρ 1.21 + 2ρ 1 2 .2 + 2ρ 2+ + 2ρ 2-+ 2ρ 31 + 4ρ 1.3 + 2Id + h 13 t 6 ρ 2+ + ρ 2-+ ρ 31 + 2ρ 1.3 + 3Id + h 16 t 8 Id.

	Remark 2.11. Unfortunately, this method is not sufficient for determining the
	D 4 -isotypic part (see also Remark 2.3).

• c (X f ) and H • c (X o ). For the characters St and Id, we proceed exactly as in Section 2.1.1 using [11, Proposition 3.3.14] and [11, Proposition 3.3.15]. Proposition 2.10.

  {t 1 t 4 t 3 t 2 t 4 t 5 t 6 , t 1 t 3 t 4 t 2 t 4 t 5 t 6 , t 1 t 3 t 4 t 3 t 2 t 5 t 6 } H

  6 [θ 2 ] + h 7 tφ 6,25 + t 3 D 4,r + h 8 t 2 φ 15,17 + t 4 D 4,Id + h 9 t 3 φ 20,10 + h 10 t 4 φ 15,5 + h 11 t 5 φ 6,1 + h 12 t 6 Id. If we exclude St and Id, none of the characters in the principal series that appear here can appear in the cohomology of U I \X W I w 0 . From that observation one can readily deduce the contribution of the principal series to the cohomology of U I \X(w). Note that in the case of St and Id we can proceed as in Section 2.1.1. Let w = t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 and I = ∆ {t 7 }. The contribution of the principal series to the cohomology of U I \X(w) is given by h 9 St + h 10 t 2 St + φ 6,25 + φ 20,20 + h 11 t 3 φ 6,25 + φ 20,20 + φ 15,17 + φ 64,13 + h 12 t 4 φ 15,17 + φ 64,13 + φ 20,10 + φ 90,8 + h 13 t 5 φ 20,10 + φ 90,8 + φ 15,5 + φ 64,4 + h 14 t 6 φ 15,5 + φ 64,4 + φ 6,1 + φ 20,2 + h 15 t 7 φ 6,1 + φ 20,2 + Id + h 18 t 9 Id.

	Proposition 2.17.

  Proposition 2.19. Let w = t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 . The cuspidal part of the cohomology of X(w) is given by h 10 t 9/2 E 7 [i] + E 7 [-i] . 2.3.3. Cohomology of X(w). By combining Proposition 2.17 and 2.18, we obtain the Harish-Chandra restriction to E 6 of the cohomology of the variety X(w). If we compare these to the restriction of the characters in the principal Φ 14 -block b uni = {St G , Id G , φ 27,37 , φ 105,26 , φ 189,17 , φ 189,10 , φ 105,5 , φ 27,2 , D 4,1 3 . , D 4,1 2 .1 , D 4,1.2 , D 4,.3 , E 7 [i], E 7 [-i]} (and the fact that these actually occur as constituents of the cohomology) we deduce their exact contribution. Adding the cuspidal characters obtained in 2.19, we get

	i	9	10	11	12	13

Theorem 2.20. Let (G, F) be a split group of type E 7 and w be a good 14-regular element of W. The contribution to the cohomology of the Deligne-Lusztig X(w) of the principal series, the D 4 -series and the cuspidal characters coincides with the contribution of the principal Φ 14 -block, and it is given by

  = t 8 t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 .

			t 2
	1	t 3	t 4	t 5	t 6	t 7	t 8
	and choose the following 24th root of π:

w

  t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 ), Q ℓ . The latter has been computed in the previous section, whereas the cohomology of a Deligne-Lusztig variety associated to any Coxeter element c I of W I can be deduced from [20, Table7.3]:H X L I (c I ) = h 7 St + t 2 D 4,ε + t 3 (E 6 [θ] ε + E 6 [θ 2 ] ε ) + t 7/2 (E 7 [i] + E 7 [-i]) + h 8 tφ 7,46 + t 3 D 4,1.1 2 + t 4 (E 6 [θ] Id + E 6 [θ 2 ] Id ) + h 9 t 2 φ 21,33 + t 4 D 4,2.1 + h 10 t 3 φ 35,22 + t 5 D 4,Id + h 11 t 4 φ 35,13 + h 12 t 5 φ 21,6 + h 13 t 6 φ 7,1 + h 14 t 7 Id. Let w = t 8 t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 and I = ∆ {t 8 }. The contribution of the principal series to the cohomology of U I \X(w) is given by h 10 St + h 11 t 2 St + φ 7,46 + φ 27,37 + h 12 t 3 φ 7,46 + φ 27,37 + φ 21,33 + φ 105,26 + h 13 t 4 φ 21,33 + φ 105,26 + φ 35,22 + φ 189,17 + h 14 t 5 φ 35,22 + φ 189,17 + φ 35,13 + φ 189,10 + h 15 t 6 φ 35,13 + φ 189,10 + φ 21,6 + φ 105,5 + h 16 t 7 φ 21,6 + φ 105,5 + φ 7,1 + φ 27,2 + h 17 t 8 φ 7,1 + φ 27,2 + Id + h 20 t 10 Id. The results for the intermediate series depend whether the assumptions 2.16 and 2.21 are satisfied. If they hold, we can easily obtain: Let w = t 8 t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 and I = ∆ {t 8 }. (i) Under Assumption 2.16, the contribution of the D 4 -series to the cohomology of U I \X(w) is given by h 10 t 3 D 4,ε + D 4,1 3 . + h 11 t 4 D 4,ε + D 4,1 3 . + D 4,1.1 2 + D 4,1 2 .1 + h 12 t 5 D 4,1.1 2 + D 4,1 2 .1 + D 4,2.1 + D 4,1.2 + h 13 t 6 D 4,2.1 + D 4,1.2 + D 4,Id + D 4,.3 + h 14 t 7 D 4,Id + D 4,.3 . (ii) Under Assumption 2.21, the contribution of the E 6 -series to the cohomology of U I \X(w) is given by h 10 t 4 E 6 [θ] ε + h 11 t 5 E 6 [θ] ε + E 6 [θ] Id + h 12 t 6 E 6 [θ] Id and h 10 t 4 E 6 [θ 2 ] ε + h 11 t 5 E 6 [θ 2 ] ε + E 6 [θ 2 ] Id + h 12 t 6 E 6 [θ 2 ] Id .

	Together with Theorem 2.20, this is enough to determine the contribution of the principal series: Proposition 2.22. Proposition 2.23. 2.4.2. Cuspidal characters. The group G has several cuspidal unipotent char-acters, denoted in [5] by E 8 [±i], E 8 [±θ], E 8 [±θ 2

  8 t 7 t 6 t 4 t 5 t 2 t 4 t 3 t 1 , t 8 t 7 t 6 t 5 t 4 t 2 t 4 t 3 t 1 , t 8 t 7 t 6 t 5 t 4 t 5 t 2 t 3 t 1 , t 8 t 7 t 6 t 5 t 5 t 2 t 4 t 3 t 1 .

Proposition 2.25. Let

  w = t 8 t 7 t 6 t 5 t 4 t 5 t 2 t 4 t 3 t 1 . The cuspidal part of the cohomology of X(w) is given byh 10 t 5 E 8 [i] + E 8 [-i] + h 11 t 5 E 8 [-θ] + E 8 [-θ 2 ]. We summarize the results obtained in this section. The unipotent characters in the principal Φ 24 -bloc b are given byb uni = Id G , St G ,φ 35,74 , φ 160,55 , φ 350,38 , φ 448,25 , φ 350,14 , φ 160,16 , φ 35,2 , D 4,φ ′′

			2,16	,
	D 4,φ ′′ 8,9	, D 4,φ 12,4 , D 4,φ ′ 8,3	, D 4,φ ′ 2,4

2.4.3. Cohomology of X(w).

  Brauer tree of the principal Φ 24 -block of E 8

				E 6 [θ] φ ′′ 1,3
			14
					E 6 [θ] φ 2,2
				13
	D 4,φ ′ 2,4	D 4,φ ′ 8,3	D 4,φ 12,4	D 4,φ ′′ 8,9	D 4,φ ′′ 2,16 E 6 [θ] φ ′ 1,3 12 11	E 8 [i]	E 8 [-θ 2 ] 9	φ 35,2	Id
	19	18	17	16	15					8	10
					20	1,3 E 6 [θ 2 ] φ ′ 23	E 8 [-i]	1
				21
			22		E 6 [θ 2 ] φ 2,2
				E 6 [θ 2 ] φ ′′ 1,3
						Figure 4:

SOME PARTICULAR CASES

Acknowledgements

I am indebted to Jean Michel for introducing me to the methods of [11] and [9] which I have used throughout this paper. Most of this work was carried out while I was visiting him and François Digne. I would like to thank them for many valuable comments and suggestions.

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


Olivier Dudas

Cohomology of Deligne

published or not. The documents may come

Introduction

Let G be a connected reductive algebraic group over F = F p with an F qstructure associated to a Frobenius endomorphism F. Let ℓ be a prime number different from p and b be a unipotent ℓ-block of G F = G(F q ). When ℓ is large, the defect group of b is abelian, and the geometric version of Broué's conjecture predicts that the cohomology of some Deligne-Lusztig variety should induce a derived equivalence between b and its Brauer correspondent [START_REF] Broué | Zyklotomische Heckealgebren[END_REF].

When the centraliser of the defect group of b is a torus, then in [START_REF] Broué | Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées[END_REF] Broué and Michel identified which specific class of Deligne-Lusztig varieties should be considered. They correspond to good d-regular elements or equivalently to droots of π = w 2 0 in the Braid monoid. In a recent work [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF], Digne and Michel introduced the notion of d-periodic element to generalise this to the parabolic setting. If b is a unipotent Φ d -block, then it is to be expected that there exists a d-periodic element (I, w) such that the corresponding parabolic Deligne-Lusztig variety X(I, wF) is a good candidate for inducing the derived equivalence predicted by Broué's conjecture. Furthermore, Chuang and Rouquier conjectured in [START_REF] Chuang | Calabi-Yau algebras and perverse Morita equivalences[END_REF] that this equivalence is perverse, with a perversity function that has recently been conjectured by Craven in [START_REF] Craven | On the cohomology of Deligne-Lusztig varieties[END_REF]. Surprisingly, it can be expressed by a function C d depending only on the generic degrees of the corresponding characters.

If we restrict our attention to the characteristic zero then we obtain a conjectural explicit description of the unipotent part of the cohomology of X(I, wF). The fundamental property that we derive from Broué's conjecture is that the cohomology groups of X(I, wF) are mutually disjoint. More precisely, it can be formulated as follows:

Conjecture 1. Any d-cuspidal pair is conjugate to a pair (L wF I , χ) where (I, w) is a d-periodic element. Moreover, if F χ is the corresponding Q ℓ -local system on X(I, wF), then (I, w) can be chosen such that (i) The Q ℓ G F -modules H i X(I, wF), F χ are mutually disjoint.

(ii) If ρ is a irreducible unipotent constituent of H i X(I, wF), F χ then ρ lies in the Φ d -block associated to χ and i = C d (deg ρ/ degχ).

In addition, the endomorphism algebra End G F H • (X(I, wF), F χ ) should be endowed with a natural structure of d-cyclotomic Hecke algebra. Let us note the following important consequence of this property: the eigenvalue of any sufficiently divisible power F m of F on ρ should be q m (a ρ +A ρ -a χ -A χ )/d .

The choice of a specific d-periodic element in this conjecture is not very relevant: it is conjectured that any other d-periodic element (I, w ′ ) can be obtained from (I, w ′ ) by cyclic shifts, so that the cohomology of the corresponding varieties are isomorphic. This has already been proven when I = ∅ and F acts trivially on W (see [9, Remark 7.4]).

When d = 1, the unipotent blocks correspond to the usual Harish-Chandra series. In particular, when (G, F) has type A, there is a unique unipotent block and it contains all the unipotent characters. The purpose of this paper is to show that from the cohomology of X(π) one can actually deduce all the other interesting cases (see Corollary 3.2):

Theorem. For groups of type A, Conjecture 1 holds whenever it holds for d = 1, that is for X(π).

Let us emphasize that Conjecture 1 is known to be true only in a very small number of cases, namely when d = h is the Coxeter number by Lusztig [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF], for groups of rank 2 by Digne, Michel and Rouquier [START_REF] Digne | Cohomologie des variétés de Deligne-Lusztig[END_REF] and when d = n for A n and d = 4 for D 4 by Digne and Michel [START_REF] Digne | Endomorphisms of Deligne-Lusztig varieties[END_REF]. Therefore this theorem represents a very important step towards a proof of the geometric version of Broué's conjecture.

Even though this result depends on the conjectural description of X(π), one can give an effective proof of Conjecture 1 for principal Φ d -blocks when d > (n + 1)/2. In that case the defect group is cyclic, and the modular representation theory of the block is fully understood. We will address this problem in a subsequent paper, where we will compute the cohomology of Z ℓ of the corresponding Deligne-Lusztig variety.

To give a flavour of the proof of the main theorem, recall that for groups of type A, we have by [START_REF] Broué | Generic blocks of finite reductive groups[END_REF] a combinatorial description of the Deligne-Lusztig induction associated to X(I, wF). In terms of partitions, it corresponds to adding a certain number of d-hooks. By transitivity, one can decompose X(I, wF) by means of simpler varieties X n,d , each of which corresponds to adding a single dhook to a partition. Now, using the methods developed in [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF] one can compute the cohomology of (some quotient of) X n,d in terms of X n-1,d and X n-1,d-1 (see Theorem 2.1), providing an inductive argument to tackle Conjecture 1.

Note finally that Theorem 2.1 can be generalised to many other situations in type B, C and D. However, two main problems arise: firstly, the limit case is either X(w 0 ) or X(π) and does not contain all the unipotent characters. Secondly, the methods in [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF] work obviously for non-cuspidal unipotent characters only.

We can obtain partial results on the principal series in that situation, which we believe are too coarse to be mentioned in this paper.

Parabolic varieties in type A

Throughout this paper, G will denote any connected reductive algebraic group of type A n over F = F p . We will consider a Frobenius endomorphism F : G -→ G defining a standard F q -structure on G. Since we will be interested in unipotent characters only, we will not make any specific choice for (G, F) in its isogeny class. If H is any F-stable subgroup of G we will denote by H = H F the associated finite group.

The Weyl group W of G is the symmetric group S n and its Braid monoid B + is the usual Artin monoid. It is generated by a set S = {s 1 , . . ., s n } corresponding to simple reflections s 1 , . . ., s n of W. Following [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF], we define for 1

≤ d ≤ n + 1 v d = s 1 s 2 • • • s n-⌊ d 2 ⌋ s n s n-1 • • • s ⌊ d+1 2 ⌋ and J d = s i | ⌊ d+1 2 ⌋ + 1 ≤ i ≤ n -⌊ d 2 ⌋ ⊂ S.
We are interested in computing the cohomology of the variety

X n,d = X(J d , v d F)
with coefficients in any unipotent local system. Note that for d > 1, the element v d is reduced so that we can work with the variety X(J d , v d F). By [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF]Lemma 11.7 and 11.8], the pair (J d , v d ) is d-periodic so that it makes sense to study the cohomology of X n,d . Recall from [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF] that a d-periodic element is any pair (I, b) with

I ⊂ S and b ∈ B + such that bF(b) • • • F d-1 (b) = π/π I where π = w 2
0 is the generator of the pure Braid group. It has been shown in [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF] that this forces bF to normalise I. Note that when d ≤ (n + 1)/2, v d is not maximal in the sense that it is not extendable to a dth root of π/π I for a proper subset I of J d . However, it can still be used to associate to any unipotent block a "good" parabolic Deligne-Lusztig variety. Before making any precise statement, we shall briefly recall the combinatorial objects that we will use.

1.1. Φ d -blocks of G. The unipotent characters of G are labeled by the partitions of n+1. If λ is such a partition, we will denote by χ λ the corresponding character, with the convention that χ (1,1,...,1) = St G is the Steinberg character of G. We shall also fix a representation V λ over Q ℓ of character χ λ . For 1 ≤ d ≤ n + 1, the pair (L J d , v d F) represents a d-Levi subgroup of G. From [START_REF] Broué | Generic blocks of finite reductive groups[END_REF], we know how to express the d-Harish-Chandra induction in terms of combinatorics of partitions. To fix the notation, let µ be a partition of n + 1 -d and X = {x 1 < x 2 < • • • < x s } be a β-set associated to µ. We may and we will assume that X is big enough, so that it contains {0, 1,

• • • , d -1}. Let X ′ be the subset of X defined by X ′ = {x ∈ X | x + d ∉ X }.
It represents the possible d-hooks that can be added to µ. For x ∈ X ′ we will denote by µ * x the partition of n + 1 which has (X {x}) ∪ {x + d} as a β-set.

We fix an F-stable Tits homomorphism t : B + -→ N G (T). By [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF] the variety X n,d has an étale covering X

n,d = X(J d , v d F) with Galois group L t(v d )F J d . Since (L J d , t(v d )F) is a split group of type A n-d , the partition µ defines a unipotent local system F µ on X n,d such that H • c (X n,d , F µ ) and H • c ( X n,d , Q ℓ ) χ µ are isomorphic.
Then we deduce from [START_REF] Broué | Generic blocks of finite reductive groups[END_REF]Section 3.4] that there exist signs ε x = ±1 such that the d-Harish-Chandra induction of χ µ is given by

R G L J d (χ µ ) = (-1) i H i c (X n,d , F µ ) = x∈X ′ ε x χ µ * x .
In particular, the d-Harish-Chandra restriction of χ λ ∈ IrrG is non-zero until we reach the d-core ν of λ, which corresponds to a d-cuspidal character χ ν . The unipotent characters in the Φ d -block of G containing χ λ are all the characters that can be obtained by successive d-inductions from χ ν . They correspond to partitions of n + 1 that have ν as a d-core.

1.2.

A parabolic variety associated to a Φ d -block. The cohomology of the variety X n,d induces to a minimal d-induction since there is no d-split Levi between (L J d , t(v d )F) and (G, F). By transitivity, one can form a Deligne-Lusztig variety X(I, w) associated to the d-cuspidal character χ ν . Let n + 1 -ad be the size of ν and consider for i = 1, . . ., a the pairs (J

(i) d , v (i) d ) where (J (0) d , v (0) d ) = (J d , v d ) and (J (i+1) d , v (i+1) d
) is the analogue of the pair (J d , v d ) for the split group

(L J (i) d , t(v (i) d • • • v (1) d )F) of type A n-id .
Then one can readily check that the pair Proposition 8.26] the associated Deligne-Lusztig variety X(I, wF) is isomorphic to the following almalgamated product X(J (1) d , v (1) d F)

(I, w) = (J (a) d , v (a) d • • • v (1) d ) is d-periodic. By [9,
× L t(v (1) d )F J (1) d • • • × L t(v (a-1) d •••v (1) d )F J (a-1) d X L J (a-1) d (J (a) d , v (a) d t(v (a-1) d • • • v (1) d )F).
Now each variety in this decomposition corresponds to a variety X n-id,d for some i = 0, . . ., a -1. Since the cohomology of the latter with coefficients in a unipotent local system depends only on the isogeny class of the group (here, the split type

A n-id ) we obtain RΓ c X(I, wF , F ν ) ≃ RΓ c ( X n,d , Q ℓ ) ⊗ A n-d • • • ⊗ A n-(a-1)d RΓ c (X n-(a-1)d,d , F ν ). (1.1)
Consequently, if we believe that the vanishing property in Conjecture 1 holds for the cohomology of X(I, w), then for any partition

µ of n + 1 -d, the graded G-module H • c (X n,d , F µ
) should be multiplicity-free.

1.3. Craven's formula in type A. Conjecturally, the unipotent character χ λ is a constituent of only one cohomology group of RΓ X(I, wF , F ν ). Craven proposed in [START_REF] Craven | On the cohomology of Deligne-Lusztig varieties[END_REF] a formula which gives the degree of this cohomology group in terms of d and the generic degree of χ λ and χ µ . More precisely, he considers a function C d on some set of enhanced cyclotomic polynomials and conjectured that

χ λ ; H i (X(I, w), F ν ) G = 0 ⇐⇒ i = C d (deg χ λ ) -C d (deg χ ν ). (1.2)
Let us recall the definition of C d : assume that

P ⊂ Q[x]
is a polynomial such that the non-zero roots z 1 , . . ., z m (written with multiplicity) of P are all roots of unity. Let us denote by d • (P) the degree of P and by v(P) its valuation, that is the degree of x d • (P) P(x -1 ). Then Craven's function C d is defined by

C d (P) = 1 d d • (P) + v(P) + # i = 1, . . ., m | Arg z i < 2π/d - 1 2 # i = 1, . . ., m | z i = 1 .
Here, the argument Arg z of a non-zero complex number z is taken in [0; 2π

(PQ) = C ζ (P) + C ζ (Q).
For groups of type A, the degree deg χ λ of the unipotent character χ λ is explicitely known (see for example [5, Section 13]). It is a polynomial in q of degree A λ and valuation a λ and no factors of the form (q -1) can appear. In particular, Craven's function can be written

C ζ (deg χ λ ) = 2π Argζ a λ + A λ + # i = 1, . . ., m | Arg z i < Arg ζ
where z 1 , . . ., z m are the roots with multiplicity of the polynomial deg χ λ . Note that with this description it is already not obvious that the rational number on the right-hand side of 1.2 is actually an integer.

Since C d is additive, formula 1.2 together with the quasi-isomorphism 1.1 suggests that the partition ν should not be necessarily a d-core. In the case of an elementary d-induction (when a = 1) we can write everything explicitely using [7, Proposition 9.1]; the second equality follows from an easy calculation: Lemma 1.3. Let µ be a partition with corresponding β-set X that we assume to be large enough. For x ∈ X ′ , we have

C d (deg χ µ * x )-C d (deg χ µ ) = 2 n+1-d -x+#{ y ∈ X | y < x} +#{ y ∈ X | x < y < x+d} and a µ * x + A µ * x -a µ -A µ = d(n -d + s -x).
These integers give conjecturally the degree of the cohomology group of X n,d in which χ µ * x will appear, as well as the corresponding eigenvalue of the Frobenius. Since we will work with the cohomology with compact support, we shall rather work with the integers

π d (X , x) = 2 n + x -#{ y ∈ X | y < x} -#{ y ∈ X | x < y < x + d} and γ d (X , x) = n + 1 + x -s.
They are readily deduced from the previous ones by taking into account the dimension of X n,d (which is equal to ℓ(v d ) = 2n + 1 -d). Now Conjecture 1 can be deduced from the following:

Conjecture 1.4. Let n ≥ 1 be a positive integer and 1 ≤ d ≤ n + 1. Let µ be a partition of nd + 1 and X be its β-set, assumed to be large enough. Then

RΓ c X n,d , F µ ≃ x∈X ′ V µ * x [-π d (X , x) ⊗ Q ℓ γ d (X , x)
as a complex of G × 〈F〉-modules.

The purpose of this paper is to prove that this conjecture holds for any d whenever it holds for d = 1. As a byproduct, we shall deduce the cohomology of parabolic Deligne-Lusztig varieties associated to any unipotent block from the knowledge of the cohomology of X(w 2 0 ).

Decomposition of the quotient

The group (L J d , vd F) is split of type A n-d , therefore the unipotent representations of the corresponding finite group are labelled by partitions µ of nd + 1.

To such a partition one can associate a unipotent Q ℓ -local system F µ on X n,d .

From [START_REF] Broué | Generic blocks of finite reductive groups[END_REF] we know that the irreducible constituents of the virtual character (-1) i H i c (X n,d , F µ ) correspond to the partitions of n + 1 obtained by adding a d-hook to µ. The restriction to S n of the corresponding irreducible representation corresponds to a partition obtained by • either restricting the hook (usually in two different ways),

• or restricting µ.

The main result of this section gives a geometric interpretation of this phenomenon.

Theorem 2.1. Assume that d ≥ 2. Let I = {s j | 1 ≤ j ≤ n -1}. Let µ be a partition of n -d + 1 and {µ ( j) } be the set of partitions of n -d obtained by restricting µ. Then there is a distinguished triangle in D b (Q ℓ L I × 〈F〉-mod) RΓ c (G m ×X n-1,d-1 , Q ℓ ⊗F µ ) -→ RΓ c (X n,d , F µ ) U I -→ RΓ c X n-1,d , F µ ( j) [-2](1)
Remark 2.2. From [1, Proposition 1.1] we can deduce that the cohomology of a Deligne-Lusztig variety with coefficients in a unipotent local system depends only on the type of (G, F). Therefore there is no ambiguity in the statement of the theorem.

We will use the results in [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF] to compute the quotient of X(J d , vd F) by the finite group U I . Recall that X n,d = X(J d , v d F) can be decomposed into locally closed P I -subvarieties X x , where x is a I-reduced-J d element of W. In our situation, at most two pieces will appear: Lemma 2.3. Assume that 2 ≤ d ≤ n. The variety X x is non-empty if and only if x is one of the two following elements:

(1) x 0 = s n s n-1 • • • s 1 (2) x 1 = s n s n-1 • • • s n-⌊ d 2 ⌋+1 .
Proof. For simplicity, we shall denote [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF] that the variety X x is non-empty if and only if there exists y = y 1 • • • y r ∈ W J and an xdistinguished subexpression γ of yw such that the products of the elements of γ lies in (W I ) x . We first observe that for i ∉ {1, n + 1} we have

a = ⌊ d+1 2 ⌋ and b = n -⌊ d 2 ⌋ so that w = v d = s 1 • • • s b s n • • • s a and J = J d = {a + 1, . . ., b}. If x is a I-reduced-J element, then x = s n s n-1 • • • s i with b + 1 ≤ i ≤ n + 1 or 1 ≤ i ≤ a. Recall from
(W I ) x = 〈s 1 , . . ., s i-2 , s i s i-1 s i , s i+1 , . . ., s n 〉.
Now, since x is reduced-J, the subexpression γ is the concatenation of (y 1 , . . ., y r ) and an x y-distinguished subexpression γ of w.

If i > b + 1 or 2 ≤ i ≤ a, the group W J is included in (W I ) x .
Therefore the product of the elements of γ must lie in (W I ) x . We shall distinguish two cases:

Case (1). We assume that i > b + 1. In that case x commutes with any element of W J , so that γ is an yx-distinguished subexpression of w. Then • if x is trivial (that is if i = n+1), then any y-distinguished subexpression of w contains necessarily s n and hence cannot produce any element of (W I ) x = W I ;

• if x is non-trivial then i ≤ n, and a subexpression of w lies in (W I ) x if and only if it does not contain s i or s i-1 . However, such a subexpression will never be yx-distinguished since for all v in W I we have vxs i-1 > vx.

We deduce that the variety X x is empty in this case.

Case (2). We assume that 2 ≤ i ≤ a. The subexpression γ is x yx-distinguished.

Since i ≤ a, we have x W J = W a,...,b-1 . For j < i -1, we have xs j = s j x; moreover, xs i-1 is I-reduced, so that γ should start with (s 1 , . . ., s i-1 ). In that case, the product of the elements of γ cannot belong to (W I ) x . Indeed, a subexpression of

s i-1 s i • • • s b s n • •
• s a starting with s i-1 will never give an element of (W I ) x , the only non-trivial situation being the case a = i:

• with the notation in [10, Section 2.1.2] we have s a-1 s a+1

• • • s b s n • • • s a = s a+1 • • • s b s n • • • s a+1 s a-1 s a and neither s a-1 nor s a-1 s a belongs to (W I ) x ; • s a-1 s a s a+1 • • • s b s n • • • s a = (s a-1 s a s a-1 ) s a-1 s a+1 • • • s b s n • •
• s a and we are back to the previous case.

This forces the variety X x to be empty.

Proof of the Theorem. From the previous lemma we deduce that X(J d , v d F) decomposes as a disjoint union X(J d , v d F) = X x 0 ∪ X x 1 with X x 0 being open. Using [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF] we shall now determine the cohomology of the quotient of each of these varieties by U I . Throughout the proof, we will denote I = {s 2 , . . ., s n } ⊂ S the conjugate of I by w 0 .

When x = x 0 = w I w 0 and d > 2 we are in the situation of [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF]Proposition 3.4]. Indeed, v d = s 1 w ′ with w ′ ∈ W 2,...,n and s 1 commutes with W J d ⊂ W 3,...,n-1 so that we obtain

RΓ c U I \ X x 0 /N, Q ℓ ≃ RΓ c G m × X L I (K x 0 , vF) / ẋ0 N ′ , Q ℓ with v = x 0 w ′ and K x 0 = I ∩ x 0 Φ J d = x 0 J d .
For simplicity, we shall rather consider the conjugate by x 0 of the right-hand side Recall that N and N ′ are normal subgroup of L J d and are both contained in T. In particular, any unipotent character of

L wF J d (resp. of L w ′ F J d
) is trivial on N (resp. N ′ ). Consequently, for any unipotent χ of L wF J d we obtain the following quasi-isomorphism of complexes of L I × 〈F〉-modules:

RΓ c U I \ X x 0 , Q ℓ χ ≃ RΓ c G m × X L I ( x 0 J d , vF), Q ℓ ẋ0 χ .
Finally, we observe that the varieties X L I ( x 0 J d , vF) and X n-1,d-1 have the same cohomology with coefficients in any unipotent local system. Indeed, if we denote (s 1 , . . ., s n-1 ) by (t 1 , . . ., t n-1 ) if d is odd or by (t n-1 , . . ., t 1 ) if d is even, then we have

v = t 1 t 2 • • • t n-1-⌊ d-1 2 ⌋ t n-1 ṫn-2 • • • t ⌊ d 2 ⌋
which corresponds to the element v d-1 in the Weyl group W I = 〈t 1 , . . ., t n-1 〉 of type A n-1 .

When x = x 0 and d = 2, we can write v 2 = ww ′ with w = s n s n-1 • • • s 2 and w ′ = s 1 s 2 • • • s n = s 1 w ′′ so that X n,2 ≃ X {s 2 , . . ., s n-1 }, ww ′ F . Moreover, via this isomorphism we have X x 0 ≃ y∈W X (x 0 ,y) .

We claim that X (x 0 ,y) is empty unless y ∈ W I w 0 W J ′ 2 where J ′ 2 = J w 2 = {s 3 , s 4 , . . ., s n }. The piece X (x 0 ,y) consists of pairs (px 0 P J 2 , p ′ yP J ′ 2 ) with p, p ′ ∈ P I such that p -1 p ′ ∈

x 0 P J 2 wP J ′ 2 y -1 and p ′-1 F p ∈ yP J ′ 2 w ′ P J 2 x -1 0 .
In particular, if X (x 0 ,y) is non-empty then the double coset P I yP J ′ 2 has a non-trivial intersection with x 0 Bw. But x 0 = w I w 0 is reduced-I so that ℓ(x 0 w) = ℓ(x 0 ) + ℓ(w) and x 0 Bw ⊂ P I w 0 P J ′ 2 with K x 0 = x 0 J 2 , v = x 0 w and v ′ = x 0 w ′′ . If we denote (s 1 , . . ., s n-1 ) by (t n-1 , . . ., t 1 ) we obtain x 0 J 2 = {t 2 , . . ., t n-1 } and vv ′ = t 1 • • • t n-1 t n-1 • • • t 1 so that the pair (K x 0 , vv ′ ) corresponds to (J 1 , v 1 ) in the Weyl group W I of type A n-1 . As before, N and N ′ do not play any role if we consider the unipotent part of the previous quasiisomorphism.

When x = x 1 we use [11, Proposition 3.2]: the conjugate of v d by x 1 is v = x 1 v d x -1 1 = s 1 s 2 • • • s n-1-⌊ d 2 ⌋ s s n-2 • • • s ⌊ d+1 2 ⌋
which corresponds exactly to the element v d in W I . We can therefore identify the cohomology of the varieties X L I (K x 1 , vF) and X n-1,d with coefficients in any unipotent local system (see Remark 2.2). The group

P I ∩ x 1 L J d is a vF-stable parabolic subgroup of x 1 L J d and L K x 1 = L I ∩ x 1 L J d is a stable Levi complement.
Therefore it makes sense to consider the Harish-Chandra restriction

* R J K χ of any unipotent character χ L vd F J d ≃ ( x 1 L J d ) vF to L vF K x 1
. From [11, Proposition 3.2] (see also [START_REF] Dudas | Quotient of Deligne-Lusztig varieties[END_REF]Remark 3.12]) we deduce the following quasi-isomorphism

RΓ c U I \ X x 1 , Q ℓ χ [2](-1) ≃ RΓ c X L I (K x 1 , vF), Q ℓ * R J K χ .
Let µ be a partition of nd + 1. The cohomology of the variety X n,d with coefficients in the local system F µ is given by

RΓ c (X n,d , F µ ) ≃ RΓ c X(J d , v d F), Q ℓ χ µ
where χ µ is the unipotent character of L vd F J d corresponding to the partition µ. Since (L K x 1 , vF) is a split group of type A n-d-1 , the Harish-Chandra restriction of χ µ from L ẇF

J d ≃ ( x 1 L J d ) vF to L vF K x 1
is the sum of the χ µ i 's where the µ i 's are the partitions of nd obtained by restricting µ. With with description, we get the following isomorphisms in

D b (Q ℓ L I × 〈F〉-mod) RΓ c U I \ X x 0 , Q ℓ χ µ ≃ RΓ c G m × X n-1,d-1 , Q ℓ ⊗ F µ and RΓ c U I \ X x 1 , Q ℓ χ µ ≃ RΓ c X n-1,d , F µ i [-2](1).
We conclude using the distinguished triangle associated to the decomposition X n,d = X x 0 ∪ X x 1 in which X x 0 is open.

Cohomology over Q ℓ

We have just seen how to relate the Harish-Chandra restriction of the cohomology of X n,d to the cohomology of smaller parabolic Deligne-Lusztig varieties. We shall now explain how this strategy provides an inductive method for a thorough determination of the cohomology of X n,d with coefficients in any unipotent local system. The main result in this section gives an inductive strategy towards a proof of Conjecture 1: Note that we already know from [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] that Conjecture 1.4 holds in the Coxeter case, corresponding to (n, n + 1). Therefore d = 1 is the only limit case. But π = w 2 0 is a maximal 1-periodic element in the sense of [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF] and in that specific case, a general conjecture for the cohomology has been already formulated in [START_REF] Broué | Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées[END_REF]: a unipotent character χ λ can appear in H i c (X(π)) for i = 4ν G -2A λ only, where ν G is the number of positive roots. An important consequence of Theorem 3.1 is that knowing the cohomology of X(π) is sufficient for determining all the other interesting cases: Corollary 3.2. For groups of type A, Conjecture 1 holds for any d ≥ 1 as soon as it holds for d = 1.

Proof. Assume that Conjecture 1 holds for d = 1, that is for the variety X(π). Let

I = J 1 = {s 2 , . . ., s n } and b = v 1 = s 1 • • • s n s n • • • s 1 . By [9, Proposition 8.26] we have RΓ c X(π), Q ℓ ≃ RΓ c X(I, bF), Q ℓ ⊗ Q ℓ L t(b)F I RΓ c X(π I ), Q ℓ .
Since the cohomology of X(π I ) contains all the unipotent characters of L t(b)F I , we deduce that for any partition µ of n, the groups H i c (X n,1 , F µ ) are submodules of the cohomology groups of X(π). Consequently, they are disjoint as soon as Conjecture 1 holds for X(π). Since we have assumed that it holds also for X(π I ) we have actually

H i c X(π), Q ℓ ≃ µ-n H i-4ν L I +2A µ c X n,1 , F µ (2ν L I -a µ -A µ ) (3.3)
as a G × 〈F〉-module. Now, the alternating sum of the cohomology groups of X n,1 represents the Deligne-Lusztig induction from L t(b)F I ≃ L I to G. Therefore a character χ λ appear in H • c (X n,1 , F µ ) if and only if µ is the restriction of λ, or equivalently, if λ is obtained from µ by adding a 1-hook. This, together with 3.3 and Lemma 1.3 proves that Conjecture 1.4 holds for X n,1 , and therefore for any variety X n,d by 3.1. We use 1.1 to conclude.

Proof of the Theorem. Let X be a β-set associated the partition µ of nd +1. We can always assume that it contains {0, 1, . . ., d -1}. The partitions µ ( j) 's of nd which are obtained by restricting µ can be associated to the following β-set:

X ( j) = {x ( j) 1 < • • • < x ( j) s } with x ( j) i = x j -1 if i = j;
x i otherwise.

Let I = {s 1 , . . ., s n-1 }. By Theorem 2.1, the Harish-Chandra restriction of the cohomology of X n,d can be fitted into the following distinguished triangle:

RΓ c (G m ×X n-1,d-1 , Q ℓ ⊗F µ ) -→ RΓ c (X n,d , F µ ) U I -→ RΓ c X n-1,d , F µ ( j) [-2](1)
Now, if we assume that Conjecture 1.4 holds for both (n -1, d -1) and (n -1, d), the complexes on the left and right-hand side are completely determined. Let us examine the different eigenvalues of F that can appear:

(a) on C = RΓ c (G m × X n-1,d-1 , Q ℓ ⊗ F µ )
, the eigenvalues of F are q n+x-s and q n+1+x-s with x ∈ X such that x + d -1 ∉ X . The character of the corresponding eigenspace is χ λ where λ is the partition of n obtained by adding to µ a (d -1)-hook represented by x; 1), the eigenvalues of F are q n+1+x-s where x ∈ X ( j) is such that x + d ∉ X ( j) . The character of the corresponding eigenspace is χ λ where λ is the partition of n obtained by adding to µ ( j) a d-hook represented by x.

(b) on D ( j) = RΓ c (X n-1,d , F µ ( j) )[-2](
We shall now determine H • c (X n,d , F µ ) U I by studying each eigenspace of F separately. For x a positive integer, we can separate the following cases:

Case (1). Assume first that x ∈ X and x + d ∉ X . Let λ = µ * x be the partition of n + 1 obtained by adding to µ a d-hook from x. We want to prove that the

q n+1-s+x -eigenspace of F on H • c (X n,d , F µ ) U I is non-zero in degree π d (X , x) only
and that its character is the Harish-Chandra restriction of χ λ . By (a), the q n+1+x-s -eigenspace of F on C will produce non-zero representations in the following two cases:

• if x + d -1 ∉ X , then one obtains a character associated to the β-set (X {x}) ∪ {x + d -1} and it is concentrated in degree

2 + π d-1 (X , x) = 2 + 2 n -1 + x -#{ y ∈ X | y < x} -#{ y ∈ X | x < y < x + d -1} = 2 n + x -#{ y ∈ X | y < x} -#{ y ∈ X | x < y < x + d -1} = π d (X , x).
• if x + 1 ∈ X , then the corresponding β-set is (X {x + 1}) ∪ {x + d} and the associated character appears in degree 1 + π d-1 (X , x + 1) only. But we have

π d-1 (X , x + 1) = 2 n + x -#{ y ∈ X | y < x + 1} -#{ y ∈ X | x + 1 < y < x + d} = 2 n + x -1 -#{ y ∈ X | y < x} -#{ y ∈ X | x < y < x + d} -1 = π d (X , x) -1.
On the other hand, the q n+1+x-s -eigenspace of F on D ( j) is non-zero if and only if x ∈ X ( j) and x + d ∉ X ( j) . This happens if and only if x and x + d + 1 are different from x j . In that case, the β-set corresponding to the character of the eigenspace will be (X ( j) {x}) ∪ {x + d}. Furthermore, the degree in which this character will appear is 2 + π d (X ( j) , x), which is clearly equal to π d (X , x) in that case.

Now, the β-set Y = (X {x}) ∪ {x + d} is associated to the partition λ = µ * x. As mentioned in the beginning of Section 2, the restriction of λ is obtained by restricting the hook (usually in two different ways) or by restricting µ. In the framework of β-sets, it corresponds to decreasing specific elements of Y :

• if x + d -1 ∉ X , one can replace x + d by x + d -1 in Y and we obtain the β-set (X {x}) ∪ {x + d -1};

• if x + 1 ∈ X , one can replace x + 1 by x in Y and we obtain the β-set (X {x + 1}) ∪ {x + d};

• if x j ∈ X is different from x or from x + d + 1, and if x j -1 ∉ X , then one can replace x j by x j -1 in Y to obtain the β-set (X ( j) {x}) ∪ {x + d}. This shows that the character of the q n+1+x-s -eigenspace of

F on H • c (X n,d , F µ ) U I
is the Harish-Chandra restriction of χ µ * x . Lemma 3.4. Let λ be a partition of n+1, with n ≥ 3 and let χ be a (non-necessarily irreducible) unipotent character of G. Then the Harish-Chandra restriction of χ and χ λ are equal if and only if χ = χ λ .

Proof of the Lemma. Assume that there exists a partition ν = {ν 1 ≤ ν 2 ≤ • • • ≤ ν r } of n + 1 with ν 1 = 0 such that the difference between the Harish-Chandra restriction of χ λ and χ ν is still a unipotent character. This means that in the Young diagram of ν, any box that can be removed can be replaced to form the Young diagram of λ.

If ν = λ, this is possible only if ν 1 = ν 2 = • • • = ν r .
Let χ be a unipotent character of G which has the same Harish-Chandra restriction as χ λ . If χ = χ λ , we deduce from the previous argument that all the irreducible constituents of χ are of the form χ ν with ν = (a, a, . . ., a). This can happen if and only if n = 2 and λ = (1, 2).

When n ≥ 3, we deduce that the q n+1+x-s -eigenspace of

F on H • c (X n,d , F µ ) is actually χ µ * x . If n = 2,
then the only ambiguity concerns χ µ * x when µ * x = (1, 2). In that case, the q n+1+x-s -eigenspace can be either χ µ * x or 1 G + St G . But by [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF]Corollary 8.28], the trivial character and the Steinberg character cannot occur in the same cohomology group of X n,d as soon as the dimension of this variety is non-zero.

Case (2)

. Assume now that x ∉ X . The q n+1+x-s -eigenspace of F on D ( j) is nonzero if and only if x ∈ X ( j) and x + d ∉ X ( j) . Since x ∉ X , this forces x = x ( j) j = x j -1 and x j + d -1 ∉ X . In that case, its character corresponds to a partition with β-set (X {x + 1}) ∪ {x + d} and it appears in degree 2 + π d (X ( j) , x) only. On the other hand, the q n+1+x-s -eigenspace of F on C is non-zero if and only if x +1 ∈ X and x + 1 + d -1 = x + d ∉ X . By (a), the character of this eigenspace corresponds to a partition with β-set (X {x + 1}) ∪ {x + d}. Furthermore, it appears in degree 1 + π d-1 (X , x + 1) only. Note that in that case we have

π d-1 (X , x + 1) = 2 n + x -#{ y ∈ X | y < x + 1} -#{ y ∈ X | x + 1 < y < x + d} = 2 n + x -#{ y ∈ X | y < x} -#{ y ∈ X | x < y < x + d} -1 = π d (X , x) + 1 and π d (X ( j) , x) = 2 n -1 + x -#{ y ∈ X ( j) | y < x} -#{ y ∈ X ( j) | x < y < x + d} = 2 n -1 + x -#{ y ∈ X | y < x} -#{ y ∈ X | x < y < x + d} -1 = π d (X , x) -1.
We deduce that the q n+1+x-s -eigenspace of F on H • c (X n,d , F µ ) U I is isotypic and concentrated in two consecutive degrees. However, there are only a few unipotent characters that can have an isotypic Harish-Chandra restriction: they correspond to partitions of the form (a, a, . . ., a). Among them we can find the Steinberg character St G (with a = 1) and the trivial character 1 G (with a = n+1). But by [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF]Corollary 8.28] they have respective eigenvalues 1 and q 2n+1-d . Let us write the β-set of µ as X = {0, 1, . . ., k -1,

µ 1 + k, µ 2 + k + 1, . . ., µ r + s -1} with k ≥ d. Since x ∉ X , one must have k -1 < x < µ r + s -1 and hence d -1 ≤ n + k -s < n + 1 + x -s < n + 1 + µ r -1 ≤ 2n + 1 -d. (3.5)
We deduce that the q n+1+x-s -eigenspace of

F on H • c (X n,d , F µ ) is either zero, or
consists of two copies of the character χ λ in two consecutive degrees, where λ = (a, a, . . ., a) with 1 < a < n + 1. We shall actually prove that it is always zero, but before that we need to study the last case.

Remark 3.6. The Harish-Chandra restriction of χ λ corresponds to the partition (a -1, a, . . ., a). Therefore if the associated character appears in the cohomology of C then the β-set (X {x + 1}) ∪ {x + d} must correspond to the partition (a -1, a, . . ., a). This gives a rather strong condition on X : we will have either

X = {0, 1, . . ., k -1, x + 1, b, b + 2, b + 3, . . ., x + d, . . ., b + r} with b + 2 ≤ x + d ≤ b + r, or X = {0, 1, . . ., k -1, x + 1, x + d + 2, x + d + 3, . . ., x + d + r}.
Case (3). Finally, assume that x ∈ X and x + d ∈ X . The q n+1+x-s -eigenspace of F on D ( j) is non-zero if and only if x ∈ X ( j) and x + d ∉ X ( j) . Since x + d ∈ X , this forces x + d = x j (and therefore x j -1 ∉ X ). In that case, the character of the eigenspace corresponds to a partition with β-set (X ( j) {x}) ∪ {x + d} = (X {x}) ∪ {x + d -1}. On C , the Frobenius has a non-zero q n+1+x-s -eigenspace if and only if x + d -1 ∉ X and its character is again associated to the β-set (X {x}) ∪ {x + d -1}. This ensures that the q n+1+x-s -eigenspace of F on H • c (X n,d , F µ ) U I is isotypic. Using x+d -1 instead of x in the inequalities 3.5 yields 0 < n+1+x-s < 2n + 2 -2d and therefore the previous argument applies. We deduce that the q n+1+x-s -eigenspace of F on H • c (X n,d , F µ ) is again either zero or consists of two copies of the character χ λ in two consecutive degrees, namely π d (X , x) -1 and π d (X , x), where λ = (a, a, . . ., a) and 1 < a < n + 1.

To conclude, we need to prove that the q n+1+x-s -eigenspaces of F are actually zero whenever x ∉ X or x + d ∈ X . Let us first summarize what we have proven so far:

(1) if x ∈ X and x + d ∉ X then the q n+1+x-s -eigenspace of F on H • c (X n,d , F µ ) is χ µ * x and it appears in degree π d (X , x) only;

(2) if x ∉ X , the q n+1+x-s -eigenspace of F is zero unless x +1 ∈ X and x + d ∉ X .

In that case, it may consist of two copies of χ λ , one in degree π d (X , x) + 1 and one in degree π d (X , x) + 2, where λ = (a, a, . . ., a) with 1 < a < n + 1.

Moreover, the β-set (X {x + 1}) ∪ {x + d} must correspond to the partition (a -1, a, . . ., a) (see Remark 3.6);

(3) if x ∈ X and x + d ∈ X , the q n+1+x-s -eigenspace of F is zero unless x + d -1 ∉ X . In that case, it can only be χ λ -isotypic with λ = (a, a, . . ., a) and 1 < a < n + 1. Moreover, it is non-zero in degrees π d (X , x) -1 and π d (X , x) only, and (X {x})∪{x+d -1} must be a β-set of the partition (a-1, a, . . ., a).

Now, if we assume that Conjecture 1.4 holds for the variety X n,d+1 , then we can use the distinguished triangle

RΓ c (G m ×X n,d , Q ℓ ⊗F µ ) -→ RΓ c (X n+1,d+1 , F µ ) U I -→ RΓ c X n,d+1 , F µ ( j) [-2](1)
from Theorem 2.1 to prove that the eigenspaces of F on H • c (X n,d , F µ ) in cases [START_REF] Broué | Zyklotomische Heckealgebren[END_REF] and ( 3) are indeed zero.

Assume that x ∉ X and that there exists 1 < a < n + 1 such that the character χ λ = χ (a,a,...,a) appears twice in the q n+1+x-s -eigenspace of F on the cohomology of X n,d -that is in degrees π d (X , x) + 1 and π d (X , x) + 2. Then,

• if x -1 ∉ X , the q n+1+x-s -eigenspace of F on H • c (G m × X n,d , Q ℓ ⊗ F µ ) is χ λ -
isotypic by (2) (we have x -1 + 1 ∉ X ). Moreover, the q n+1+x-s -eigenspace of F on H • c X n,d+1 , 1) is zero since Conjecture 1.4 holds for the variety X n,d+1 . We deduce that the eigenspace on H • c (X n+1,d+1 , F µ ) U I is χ λisotypic, which is impossible since no unipotent character can have χ λ as a Harish-Chandra restriction when 1 < a < n + 1.

F µ ( j) [-2](
• if x -1 ∈ X , then since x -1 + d + 1 ∉ X , the q n+1+x-s -eigenspace of F on H • c (G m × X n,d , Q ℓ ⊗ F µ ) and H • c X n,d+1 , F µ ( j) [-2](1) can be determined as in case [START_REF] Bonnafé | Coxeter orbits and modular representations[END_REF]. It corresponds to the Harish-Chandra restriction of the partition µ * (x -1) obtained from µ by adding a (d + 1)-hook from x -1. Furthermore, they will appear in degree π d+1 (X , x -1) only, which is equal to

π d+1 (X , x -1) = 2 n + x -#{ y ∈ X | y < x -1} -#{ y ∈ X | x -1 < y < x + d} = 2 n + x + 1 -#{ y ∈ X | y < x} -#{ y ∈ X | x < y < x + d} = π d (X , x) + 2.
To these characters we have to add the contribution of χ λ and possibly of an other character χ λ ′ corresponding to λ = (a ′ , a ′ , . . ., a ′ ) (when the case (3) applies to x -1). Now, we claim that neither χ λ nor χ ′ λ can appear in the q n+1+x-s -eigenspace of F on H We deduce that the restriction of µ * (x-1) will never produce λ or λ ′ unless r = 2, b = k + 2 and d = 4 in the first case, or r = 2 and d = 2 in the second case. In these very specific cases, we have either X = {0, . . ., k-1, k+1, k+2}, which corresponds to the partition µ = (1, 1) or X = {0, . . ., k -1, k + 1, k + 4}, which corresponds to µ = [START_REF] Bonnafé | Coxeter orbits and modular representations[END_REF][START_REF] Broué | Generic blocks of finite reductive groups[END_REF]. In this situation, we get λ = (3, 3) and µ * (x-1) = (2, 2, 3). But [START_REF] Broué | Generic blocks of finite reductive groups[END_REF][START_REF] Broué | Generic blocks of finite reductive groups[END_REF] cannot be obtain by restricting µ * (x-1). This proves that the q n+1+x-s -eigenspace of F on H π d (X ,x)+2 c (X n+1,d+1 , F µ ) U I is just χ λ (plus possibly χ λ ′ ), which is impossible by the properties of the Harish-Chandra restriction.

The same argument can be adapted to deal with the case (3), if we rather look at the q n+2+x-s -eigenspace and distinguish whether x + 1 + d is an element of X or not. The details are left to the reader.

Theorem 3 . 1 .

 31 Let n ≥ 1 and 2 ≤ d ≤ n. If Conjecture 1.4 holds for (n, d + 1), (n -1, d -1) and (n -1, d) then it holds for (n, d).

  • c X n,d+1 , F µ ( j)[-2](1). Indeed the assumptions on x force X (see Remark 3.6) to be eitherX = {0, 1, . . ., k -1, k + 1, b, b + 2, b + 3, .. ., k + d, . . ., b + r} with b + 2 ≤ k + d ≤ b + r, or X = {0, 1, . . ., k -1, k + 1, k + d + 2, k + d + 3, . . ., k + d + r}. Therefore a β-set corresponding to the partition µ * (x -1) of n + 2 is either {0, 1, . . ., k -2, k + 1, b, b + 2, b + 3, .. ., b + r} with b + 2 ≤ k + d ≤ b + r, or {0, 1, . . ., k -2, k + 1, k + d, k + d + 2, k + d + 3, .. ., k + d + r}.

Acknowledgments

I would like to thank David Craven for introducing me to the results in [7] and for many stimulating discussions around the formula he discovered. I would also like to thank the Hong Kong University of Science and Technology, and especially Xuhua He, for providing me excellent research environment during the early stage of this work.

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

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