On purity theorem of Lusztig's perverse sheaves

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 ${\mathcal{H}}^i(\mathcal{L}){\mathrm{\mid }}_x$ 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})$areequalto$(\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.