Generic Character Sheaves for Parahoric Subgroups

Lusztig's theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. In this talk, we will describe a construction which extends this "depth zero" picture to give positive-depth character sheaves associated to generic data. In the simplest nontrivial case, this resolves a conjecture of Lusztig and produces conjugation-equivariant simple perverse sheaves associated to sufficiently generic multiplicative local systems whose trace-of-Frobenius function coincides with parahoric Deligne--Lusztig induction. This is joint work with R. Bezrukavnikov.