Title: 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.