Parity sheaves on graded Lie algebras

Abstract: Let $G$ be a complex, connected, reductive, algebraic group, and $\chi: \mathbb{C}^\times \to G$ be a fixed co-character that defines a grading on $\mathfrak{g},$ the Lie algebra of $G$. Let $G_0$ be the centralizer of $\chi(\mathbb{C}^\times).$ Here I will talk about $G_0$-equivariant parity sheaves on the $n$-graded piece, $\mathfrak{g}_n.$ For the first half we will spend on derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. I will define parabolic induction and restriction in graded setting. We will dive into the results of Lusztig in characteristic $0$ in the graded setting. Under some assumptions on the field $\mathbb{k}$ and the group $G$ we will recover some results of Lusztig in characteristic $0.$ These assumption together with Mautner's cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on $\bf{Z}$-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.

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