Harish-Chandra Series in Finite Unitary Groups and Crystal Graphs

Abstract

Abstract. The distribution of the unipotent modules (in nondefining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.

Categorical actions on unipotent representations of finite classical groups

“…Therefore we need to consider a slight variation of the usual Harish-Chandra theory. The following definitions and properties are taken from [36]. It is the set of irreducible consituents in the head of F k D for some k 0 (or equivalently in the socle).…”

“…The functor F corresponds to a Harish-Chandra induction from G n (q) to G n+1 (q) whereas E corresponds to the restriction. Note that only specific Levi subgroups are considered, and we must work with a variation of the usual Harish-Chandra theory (the weak Harish-Chandra theory) introduced in [36].…”

“…Our original motivation for constructing a categorical action of g on U R comes from a conjecture of Gerber-Hiss-Jacon [36] for finite unitary groups, which predicts an explicit relation between the Harish-Chandra branching graph and the crystal graph of the Fock spaces t 0 F(Q t ) when e is odd. See also [37].…”

Heisenberg algebra, wedges and crystals

“…More precisely, the study of crystals and canonical bases of the level ℓ Fock space representations F s of U ′ q ( sl e ) gives answers to several classical problems in combinatorial terms. In particular, we know that the U ′ q ( sl e )-crystal graph of F s can be categorified in the following ways: -by the parabolic branching rule for modular cyclotomic Hecke algebras [1], when restricting to the connected component containing the empty ℓ-partition, -by Bezrukavnikov and Etingof's parabolic branching rule for cyclotomic rational Cherednik algebras [23], -by the weak Harish-Chandra modular branching rule on unipotent representations for finite unitary groups [10], [3], for ℓ = 2 and s varying. In each case, the branching rule depends on some parameters that are explicitly determined by the parameters e and s of the Fock space.…”

Kleshchev multipartitions and extended Young diagrams

“…As a consequence, we obtain a proof of a generalization of a conjecture by Dipper, James and Murphy and a generalization of the LLT algorithm for arbitrary level.2010 Mathematics Subject Classification: 20C08,05E10,17B37• in the context of the representation theory of rational Cherednik algebras: one can identify the so called KZ-component of the "Cherednik crystal of irreducible representations" with these multipartitions (see for example [15,7])• in the context of the representation theory for finite reductive groups (see [11]). For l = 2, Uglov 2-partitions are known to label some unipotent modules of the unitary group, and the associated graph has a remarkable interpretation in Harish-Chandra theory [14].All these connections give strong motivations for studying theses types of multipartitions. If l = 1, the lpartitions may be identified with the partitions and the Uglov l-partitions have then a very simple definition: they are given by the e-regular partitions (that is the partition when no nonzero part are repeated e or more times.)…”