Modular generalized Springer correspondence I: the general linear group

TL;DR

This paper introduces a generalized Springer correspondence for GL(n), explicitly determines cuspidal pairs, and constructs a stratification of perverse sheaves with connections to symmetric group representations.

Contribution

It defines a new generalized Springer correspondence for GL(n), determines cuspidal pairs, and establishes a stratification with categorical equivalences.

Findings

Explicit computation of the generalized Springer correspondence for GL(n)

Identification of cuspidal pairs in the context

Construction of a stratification with symmetric group representation categories

Abstract

We define a generalized Springer correspondence for the group GL(n) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL(n) satisfying the `recollement' properties, and with subquotients equivalent to categories of representations of a product of symmetric groups.