Modular generalized Springer correspondence II: classical groups

Abstract

Abstract. We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-0 coefficients. We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for SL(n) with coefficients of arbitrary characteristic and for SO(n) and Sp(2n) with characteristic-2 coefficients.

Constructible sheaves on nilpotent cones in rather good characteristic

“…By compatibility of the (ordinary) Springer correspondence with induction (which follows e.g. from [AHJR3,Theorem 4.5]), we deduce that…”

“…Finally, suppose that G = Spin(n). The proof is similar to the case of Sp(2n), using the descriptions of Levi subgroups admitting cuspidal pairs from [AHJR3,§8] and explicit formulas for the number of elements in each series. We omit further details.…”

“…Here P is a parabolic subgroup of G with Levi factor L; the set N (L,OL,EL) G,k does not depend on the choice of P (see [AHJR3,§2.2]), and depends only on the G-conjugacy class of (L, O L , E L ).…”

“…Given ν ∈ Part(n−k(k +1)/2, ℓ), we have N G (M ν )/M ν ∼ = (Z/2Z)≀S m(ν) , where S m(ν) is a certain product of symmetric groups defined in [AHJR3,§5.4]. The irreducible k-representations of this group are labelled by a certain combinatorial set of tuples of ℓ-regular bipartitions, denoted Bipart ℓ (m(ν)).…”

“…In this paper, which follows the series [AHJR2, AHJR3, AHJR4] on the modular generalized Springer correspondence, we consider G-equivariant constructible sheaves on the nilpotent cone N G of G with coefficients in a field k of rather good characteristic. We will study not only the category of G-equivariant perverse sheaves Perv G (N G , k), as in [AHJR2,AHJR3,AHJR4], but also the G-equivariant bounded derived category D b G (N G , k). The main idea is that in rather good characteristic, it is possible to make meaningful comparisons with characteristic-0 generalized Springer theory, a subject that has been extensively studied for over thirty years since the work of Lusztig [Lu1].…”