A quantum Mirković-Vybornov isomorphism

Abstract

We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in g l N \mathfrak {gl}_N to a slice between spherical Schubert varieties in the affine Grassmannian of P G L n PGL_n (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic W-algebra and of the latter by a truncated shifted Yangian. Building on earlier work of Brundan and Kleshchev, we define an explicit isomorphism between these non-commutative algebras and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov. As a corollary, we deduce that the W-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and Ovsienko.