On purity theorem of Lusztigs 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 $mathcal{H}^i(mathcal{L})|_x$ for a simple perverse sheaf $mathcal{L}in mathcal{Q}_{alpha}$ and $xin mathbb{E}_{alpha}^{F^n}=mathbb{E}_{alpha}(mathbb{F}_{q^n})$ 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.