Affine Springer fibers, Procesi bundles, and Cherednik algebras

Abstract

Let $\mathfrak{g}$ be a semisimple Lie algebra, let $\mathfrak{t}$ be its Cartan subalgebra, and let W be the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel–Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus {\mathfrak{t}}^{\ast })∕W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimodules over rational and trigonometric Cherednik algebras.