A quantum Mirković-Vybornov isomorphism

Webster, Ben; Weekes, Alex; Yacobi, Oded

doi:10.1090/ert/536

A quantum Mirković-Vybornov isomorphism
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by Ben Webster, Alex Weekes and Oded Yacobi

Represent. Theory 24 (2020), 38-84

DOI: https://doi.org/10.1090/ert/536

Published electronically: January 16, 2020

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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 $PGL_n$ (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.

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