Abstract. We study the decomposition matrices of the unipotent ℓ-blocks of finite special unitary groups SU n (q) for unitary primes ℓ larger than n. Up to few unknown entries, we give a complete solution for n = 2, . . . , 10. We also prove a general result for two-column partitions when ℓ divides q + 1. This is achieved using projective modules coming from the ℓ-adic cohomology of Deligne-Lusztig varieties.

Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincide with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broué's abelian defect groups conjecture for unipotent ℓ-blocks at linear primes ℓ.

We study the cohomology with modular coefficients of Deligne-Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal l-block of a finite reductive group G(F_q) when the order of q modulo l is assumed to be the Coxeter number. These results include the determination of the planar embedded Brauer tree of the block (as conjectured by Hiss, L\"ubeck and Malle) and the derived equivalence predicted by the geometric version of Brou\'e's conjecture.Comment: v2: minor corrections (including the Brauer tree of 2G2

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The purpose of this paper is to discuss the validity of the assumptions (W) and (S) stated in [12], about the torsion in the modular ℓ-adic cohomology of Deligne-Lusztig varieties associated with Coxeter elements. We prove that both (W) and (S) hold except for groups of type E 7 or E 8 .

This article is the final one of a series of articles (cf. [19,18]) on certain blocks of modular representations of finite groups of Lie type and the associated geometry. We prove the conjecture of Broué on derived equivalences induced by the complex of cohomology of Deligne-Lusztig varieties in the case of Coxeter elements whenever the defining characteristic is good. We also prove a conjecture of Hiß, Lübeck and Malle on the Brauer trees of the corresponding blocks. As a consequence, we determine the Brauer trees (in particular, the decomposition matrix) of the principal ℓ-block of E 7 (q) when ℓ | Φ 18 (q) and E 8 (q) when ℓ | Φ 18 (q) or ℓ | Φ 30 (q). * The first author is supported by the EPSRC, Project No EP/H026568/1 and by Magdalen College, Oxford 1 FINITENESS OF COMPLEXES OF CHAINS 3 the assumption that the Sylow ℓ-subgroups of G F are cyclic, and that L is the centraliser of one of them, we study the generalised eigenspaces of the Frobenius on the complex of cohomology (in the derived equivalence situation, they correspond to the images of the simple modules for N G (D)). We determine their class in the stable category of G F .The third section is devoted to mod-ℓ representations of G F = G(F q ), where G is simple and the multiplicative order of q modulo ℓ is the Coxeter number of (G, F) (with a suitable modification for Ree and Suzuki groups). We show that the knowledge of the stable equivalence induced by the Coxeter Deligne-Lusztig variety, together with the vanishing results of [18], determine Green's walk around the Brauer tree of the principal block, as predicted by Hiß-Lübeck-Malle [29]. We also show how to determine the Brauer trees of the non-principal blocks. Finally, we draw the new Brauer trees for the types 2 F 4 , F 4 , E 7 and E 8 . 1 Finiteness of complexes of chains 1.1 Good algebras 1.1.1 Locally finite modules. Let k be a field. Given B a k-algebra, we denote by B-Mod the category of left B-modules, by B-mod the category of B-modules that are finite-dimensional over k and by B-locfin the category of locally finite Bmodules, i.e., B-modules which are union of B-submodules in B-mod. These are Serre subcategories of B-Mod. We denote by B-Proj (resp. B-proj) the category of projective (resp. finitely generated projective) B-modules.Given C an additive category, we denote by Comp(C) its category of complexes and by Comp b (C) its subcategory of bounded complexes. We denote by Ho(C) the homotopy category of complexes of C.Assume now C is an abelian category. Let C ∈ Comp(C) and let n ∈ Z. We put and τ ≤n C = · · · −→ C n−2 −→ C n−1 −→ ker d n −→ 0The derived category of C will be denoted by D(C). Given I a subcategory of C, we denote by D I (C) the full subcategory of D(C) of complexes with cohomology in I. We put Ho(B) = Ho(B-Mod) and D(B) = D(B-Mod). Recall that an object of D(B) is perfect if it is isomorphic to an object of Comp b (B-proj). We refer to [30, §8.1] for basic definitions and properties of unbounded derived categories. Lemma 1.1. The category D(B-locfin) is a triangulated ...

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Abstract. We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after ℓ-reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand-Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand-Graev representations attached to cuspidal unipotent classes, as defined and studied in [14].