On purity theorem of Lusztig's perverse sheaves

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.