On purity theorem of Lusztig's perverse sheaves

Abstract

Let $Q$ be a finite quiver without loops and $\mathcal{Q}_α$ be the Lusztig category for any dimension vector $α$. 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 $\mathcal{L}\in \mathcal{Q}_α$ and $x\in \mathbb{E}_α^{F^n}=\mathbb{E}_α(\mathbb{F}_{q^n})$ 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.