Quiver varieties and symmetric pairs

Li, Yiqiang

doi:10.1090/ert/522

Quiver varieties and symmetric pairs

Author: Yiqiang Li
Journal: Represent. Theory 23 (2019), 1-56
MSC (2010): Primary 16S30, 14J50, 14L35, 51N30, 53D05
DOI: https://doi.org/10.1090/ert/522
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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