Modular Generalized Springer Correspondence: an Overview

This is an overview of our series of papers on the modular generalized Springer correspondence [AHJR2, AHJR3, AHJR4, AHJR5]. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture ‘Algebraic Groups and their Representations’ Workshop honouring G. Lusztig. The material that has not appeared in print before includes some discussion of the motivating idea of modular character sheaves, and heuristic remarks about geometric functors of parabolic induction and restriction. 1. Motivation: modular character sheaves One of Lusztig’s fundamental contributions to the development of geometric representation theory was the theory of character sheaves on a connected reductive group [L3, L4, L5, L6, L7]. This theory complemented his earlier work with Deligne on the representation theory of finite reductive groups [DeL, L1], and was crucial to the later development of algorithms to compute the irreducible characters of such groups [L8, L12, L13, Sh2, Sh3]. Introductions to this theory are given in [L9, Sh1, MS]. The subsequent thirty years have seen several generalizations of character sheaves, including generalizations to wider classes of algebraic groups; some of these are surveyed in [L15], and see also [Gi, FGT, Boy, BoyD, ShS1, ShS2, ShS3]. To date, these generalizations, like Lusztig’s original theory, have used sheaves or D-modules with coefficients in a field of characteristic zero, often Q`. Thus, they mimic ordinary (characteristic-zero) representations of finite groups. It is natural to hope for a theory of modular character sheaves that mimics modular (positive characteristic) representations of finite groups. Such a theory should complement the existing modular applications of Deligne–Lusztig theory (see [GH] for a particularly relevant survey and [CE] for a general introduction and references), and could perhaps lead, for example, to a geometric interpretation of the decomposition matrix of a finite reductive group in the case of non-defining characteristic. This hope builds on recent progress in other areas of modular geometric representation theory, which may be loosely defined as the use of sheaves with characteristic-` coefficients to answer questions about representations over fields of characteristic `. This theme arose in the work of Mirković–Vilonen [MV] and Soergel [So]. Just as modular representation theory is made difficult by the failure of Maschke’s Theorem, modular geometric representation theory is made difficult by the failure of the Decomposition Theorem of Bĕılinson–Bernstein– Deligne–Gabber [BBD]. However, works such as [JMW2] indicate ways to overcome this 2010 Mathematics Subject Classification. Primary 17B08, 20G05. P.A. was supported by NSA Grant No. H98230-15-1-0175 and NSF Grant No. DMS-1500890. A.H. was supported by ARC Future Fellowship Grant No. FT110100504. D.J. and S.R. were supported by ANR Grant No. ANR-13-BS01-0001-01.