On purity theorem of Lusztig's perverse sheaves

TL;DR

This paper proves a conjecture about the purity of Frobenius eigenvalues in Lusztig's perverse sheaves for quivers, and applies this to establish the existence of certain Hall polynomials.

Contribution

It establishes the purity of Frobenius eigenvalues for Lusztig's perverse sheaves on quivers, confirming Schiffmann's conjecture and deriving Hall polynomial existence.

Findings

Frobenius eigenvalues are equal to $( ext{sqrt}(q^n))^i$ for simple perverse sheaves.

Proof of Schiffmann's conjecture on purity.

Existence of a class of Hall polynomials.

Abstract

Let $Q$ be a finite quiver without loops and $\mathcal{Q}_{\alpha}$ be the Lusztig category for any dimension vector $\alpha$. The purpose of this paper is to prove that all Frobenius eigenvalues of the $i$-th cohomology $\mathcal{H}^i(\mathcal{L})|_x$ for a simple perverse sheaf $\mathcal{L}\in \mathcal{Q}_{\alpha}$ and $x\in \mathbb{E}_{\alpha}^{F^n}=\mathbb{E}_{\alpha}(\mathbb{F}_{q^n})$ are equal to $(\sqrt{q^n})^{i}$ as a conjecture given by Schiffmann (\cite{Schiffmann2}). As an application, we prove the existence of a class of Hall polynomials.