## LSU Doctoral Dissertations

#### Degree

Doctor of Philosophy (PhD)

Mathematics

Dissertation

#### Abstract

Let $G$ be a complex, connected, reductive, algebraic group, and $\chi:\mb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mf{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $\chi(\mb{C}^\times)$. In this dissertation, we study $G_0$-equivariant parity sheaves on $\mf{g}_n$, 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\cite{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.

5-10-2021

Daniel, Sage