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Representation Theory

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Quiver varieties and symmetric pairs

Author: Yiqiang Li
Journal: Represent. Theory 23 (2019), 1-56
MSC (2010): Primary 16S30, 14J50, 14L35, 51N30, 53D05
Published electronically: January 17, 2019
MathSciNet review: 3900699
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Abstract: We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the type $A$ case, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft–Procesi row/column removal reductions.

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Additional Information

Yiqiang Li
Affiliation: Department of Mathematics, University at Buffalo, the State University of New York, Buffalo, New York 14260
MR Author ID: 828279
ORCID: 0000-0003-4608-3465

Keywords: Nakajima variety, partial Springer resolution, nilpotent Slodowy slices of classical groups, rectangular symmetry, column/row removal reduction, symmetric pairs, $\mathscr {K}$-matrix
Received by editor(s): January 15, 2018
Received by editor(s) in revised form: October 15, 2018, and November 2, 2018
Published electronically: January 17, 2019
Additional Notes: This work was partially supported by the National Science Foundation under the grant DMS 1801915.
Article copyright: © Copyright 2019 American Mathematical Society