Constructible sheaves on nilpotent cones in rather good characteristic

Abstract

We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.Comment: 26 page

Modular generalized Springer correspondence I: the general linear group

“…Consider the functors [AHR,Proposition 4.7] and [AM,Proposition 3.1], these functors restrict to exact functors…”

“…(Here γ G P is the left adjoint to the forgetful functor For , and it also restricts to an exact functor (see [AM,Proposition 3…”

Study of parity sheaves arising from graded Lie algebra

“…By [AJHR,Lemma 2.1], a prime is rather good if and only if l does not divide |A G (x)| for any x ∈ N G . The following is a part of a series of (unpublished) conjectures by C. Mautner.…”

Nilpotent centralizers and good filtrations

“…This is equivalent to requiring G to be standard in the sense of [14, §4]. It follows from [8,Theorem 1.8] and [3,Lemma 2.1] that when p is pretty good for G, it does not divide the order of G x /(G x ) • ∼ = G x red /(G x red ) • for any nilpotent element x. As an immediate consequence of Theorem 1.1 and Lemma 2.2 below, we have the following result.…”