Study of parity sheaves arising from graded Lie algebra

TL;DR

This paper investigates $G_0$-equivariant parity sheaves on graded Lie algebras arising from complex reductive groups, establishing their relation to parabolic induction of cuspidal pairs under certain conditions.

Contribution

It demonstrates that all such parity sheaves can be realized as direct summands of parabolic inductions of cuspidal pairs, extending Lusztig's results to positive characteristic.

Findings

Every parity sheaf is a direct summand of a parabolic induction of a cuspidal pair.

Results recover Lusztig's characteristic zero findings for $GL_n$.

Applicable to a broad class of reductive groups under specified assumptions.

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 $\Bbbk$ and the group $G$. The assumption on $G$ holds for $GL_n$ 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.