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.

arXiv commentThe authors find a gap in the proof of the main result

References in corpus (1)

• Canonical bases for the quantum extended Kac-Moody algebras and Hall polynomials