Centers of KLR algebras and cohomology rings of quiver varieties

Abstract

Attached to a weight space in an integrable highest weight representation of a simply-laced Kac-Moody algebra $\mathfrak{g}$, there are two natural commutative algebras: the cohomology ring of a quiver variety and the center of a cyclotomic KLR algebra. In this note, we describe a natural geometric map between these algebras in terms of quantum coherent sheaves on quiver varieties. The cohomology ring of an algebraic symplectic variety can be interpreted as the Hochschild cohomology of a quantization of this variety in the sense of Bezrukavnikov and Kaledin. On the other hand, cyclotomic KLR algebras appear as Ext-algebras of certain particular sheaves, and thus its center receives a canonical map from the Hochschild cohomology of the category. We show that this map is an isomorphism in finite type, and injective in general. We further note that the Kirwan surjectivity theorem for quivers of finite type is an easy corollary of these results. The most important property of this map is its compatibility with actions of the current algebra on both the cohomology of quiver varieties and on the Hochschild cohomology of any category with a categorical action of $\mathfrak{g}$. The structure of these current algebra actions allow us to show the desired results.