Quiver varieties and symmetric pairs

Abstract

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their anti-symplectic 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 construct geometrically an action of a twisted Yangian on torus equivariant cohomology of Nakajima varieties. In 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.

Spaltenstein varieties of pure dimension

“…We writeS e Pµ ,x in Section 1.3 asS Lie(G) e Pµ ,λ when the Jordan type of x is λ. The following result is obtained in [Li19,Corollary 8.3.4].…”

“…In the section, we recall briefly Nakajima varieties [N94, N98] and their σ variants in [Li19]. Our treatment follows closely Sections 1-4 in [Li19].…”

“…4.5. By the rectangular symmetry in [Li19], one can produce more examples from previous subsections. For example, the corresponding case in Section 4.1 for (G, P, x) is G ′ = Sp 12 (C), P ′ is chosen such that G ′ /P ′ is isomorphic to the isotropic flag varieties of (F 2 ⊂ F 5 ) with dim F i = i, and x ′ is of Jordan type (2, 3 2 , 4).…”

“…Instead we obtain the desired C * -actions in the setting of Nakajima quiver varieties [N94, N98] and their variants in [Li19], from which this paper is grown out.…”

“…This action satisfies all conditions in Theorem B and hence provides a conceptual proof of the pure dimensionality of X P x for G being a general linear group, i.e., Case 1.1(b). If G is classical, i.e., an orthogonal or symplectic group, the map π P admits a quiver description π σ,A , as a restriction of π A , in the recent work [Li19], with S e,x and S e,x realized as the fixed-point loci…”