Study of parity sheaves arising from graded Lie algebra

Abstract

Let $G$ be a complex, connected, reductive, algebraic group, and $\chi :{\mathbb{C}}^{\times }\to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $\chi ({\mathbb{C}}^{\times })$. In this paper, we study $G_0$-equivariant parity sheaves on ${\mathfrak{g}}_n$, under some assumptions on the field $\mathbb{k}$ and the group $G$. The assumption on $G$ holds for $G{L}_n$ and for any $G$, it recovers results of Lusztig in characteristic $0$. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair.

39 pages. Comments are welcome!