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A quantum Mirković-Vybornov isomorphism

Authors: Ben Webster, Alex Weekes and Oded Yacobi
Journal: Represent. Theory 24 (2020), 38-84
MSC (2010): Primary 16S80, 17B37, 20C99
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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Additional Information

Ben Webster
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

Alex Weekes
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada

Oded Yacobi
Affiliation: School of Mathematics and Statistics, University of Sydney, Camperdown, New South Wales 2006, Australia

Received by editor(s): October 6, 2017
Received by editor(s) in revised form: January 14, 2019, November 12, 2019, and December 23, 2019
Published electronically: January 16, 2020
Additional Notes: This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.
The first author was supported in part by the NSF Division of Mathematical Sciences and the Alfred P. Sloan Foundation.
The second author was supported in part by NSERC and the Ontario Ministry of Training, Colleges and Universities.
The third author was supported by the Australian Reseach Council.
Article copyright: © Copyright 2020 American Mathematical Society