Study of parity sheaves arising from graded Lie algebra

Abstract

Let G be a complex, connected, reductive, algebraic group, and χ : C × → G be a fixed cocharacter that defines a grading on g, the Lie algebra of G. Let G 0 be the centralizer of χ(C × ). In this paper, we study G 0equivariant parity sheaves on gn, under some assumptions on the field k and the group G. The assumption on G holds for GLn and for any G, it recovers results of Lusztig[Lu] in characteristic 0. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair.