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 ${\mathfrak{gl}}_N$ , to a slice between spherical Schubert varieties in the affine Grassmannian of $PG{L}_n$ (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.

v2: 48 pages. Major rewrite following referee comments. Added proof of a conjecture of Futorny, Molev, and Ovsienko that the finite W-algebra is free over its Gelfand-Tsetlin subalgebra

References in corpus (2)

• Elementary invariants for centralizers of nilpotent matrices
• Cohomology of Standard Modules on Partial Flag Varieties

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• A Demazure Character Formula for the Product Monomial Crystal
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products