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.

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

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.

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