Modular generalized Springer correspondence
by Anthony Henderson
Abstract: Lusztig's theory of character sheaves on a connected reductive algebraic group G is a geometric version of the ordinary (characteristic-0) representation theory of the corresponding finite groups G(F_q). In particular, it incorporates geometric versions of parabolic induction and restriction. The analogue of the fact that characteristic-0 representations of finite groups are semisimple is provided by the Decomposition Theorem. If one wants a similar geometric version of the modular (characteristic-l) representation theory of G(F_q), one should consider modular sheaves on G. In the modular setting, everything is harder: semisimplicity fails and so does the Decomposition Theorem. As a first step towards modular character sheaves, we studied the geometric induction and restriction functors for sheaves on the nilpotent cone of G, proving analogues of Lusztig's results on the generalized Springer correspondence. This was a joint project with Pramod Achar, Daniel Juteau and Simon Riche, now published in a series of papers.