Categorical actions on unipotent representations of finite classical groups

Abstract

We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given in [18], using Harish-Chandra induction and restriction. Then, we extend this construction to finite reductive groups of types B or C in non-defining characteristic. We show that the decategorified representation is isomorphic to a direct sum of level 2 Fock spaces. We deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces. We also obtain derived equivalences between blocks, yielding Broué's abelian defect group conjecture for unipotent ℓblocks at linear primes ℓ.

Triple crystal action in Fock spaces

“…Remark 7.8. In the context of modular representations of finite classical groups, there is a characterisation of cuspidal unipotent modules by Dudas, Varagnolo and Vasserot [6,Theorem 5.11] similar to Shan and Vasserot's characterisation of finitedimensional modules for cyclotomic Cherednik algebras. Therefore, we obtain an explicit description of these cuspidal modules which is exactly that of Theorem 7.7 with l = 2, see also [6,Section 5.5.3].…”

“…In a recent preprint [25], Losev has given a combinatorial interpretation of this categorical action, but without using Uglov's approach to the Fock space. Finally, in the context of finite classical groups, it has been shown that H plays a role in the study of unipotent modular representations, by relating the notion of weak cuspidality to the classical one, see [6,Section 5].…”

The $\mathfrak{sl}_\infty$-crystal combinatorics of higher level Fock spaces

“…This is completely analogous to Harish-Chandra series in Lie theory. In the case of finite unitary groups in positive, non-defining characteristic, the parametrization of Harish-Chandra series by cuspidal pairs is known to coincide with the parametrization of the Harish-Chandra series of a sum of type B Cherednik algebras, see [7].…”

Decomposition numbers for the principal $Φ_{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$

“…(1) First, we use Harish-Chandra induction and restriction to produce projective indecomposable modules (PIMs); (2) Second, we compute the missing PIMs (corresponding to cuspidal simple modules) using some partial information on the decomposition of Deligne-Lusztig characters on PIMs. For both of these steps, we use a truncated version of the Harish-Chandra induction and restriction coming from the categorical sl 2n -action on unipotent representations defined in [13]. The recent unitriangularity result in [4] allows us to compute the branching graph for this truncated induction, which provides the missing information for step (2) to be successful.…”