[Paper Review] Quiver varieties and symmetric pairs

Yiqiang Li|arXiv (Cornell University)|Jan 18, 2018

Algebraic structures and combinatorial models49 references3 citations

TL;DR

This paper introduces $σ$-quiver varieties as fixed-point subvarieties of Nakajima quiver varieties under symplectic and anti-symplectic involutions, establishing a geometric framework for symmetric pairs in representation theory. It proves that type A $σ$-quiver varieties realize partial Springer resolutions of nilpotent Slodowy slices for classical groups, revealing a rectangular symmetry and refining Kraft-Procesi reductions.

ABSTRACT

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their anti-symplectic cousins, which are compositions of a diagram automorphism, 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.

Motivation & Objective

• To develop a geometric representation theory for symmetric pairs $(\mathfrak{g}, \mathfrak{k})$ using fixed-point subvarieties of Nakajima quiver varieties under symplectic and anti-symplectic involutions.
• To establish a connection between the cohomology of Steinberg-type varieties in symplectic fixed-point subvarieties and the universal enveloping algebra of $\mathfrak{k}$, the Lie subalgebra in a symmetric pair.
• To construct a geometric action of the twisted Yangian on the torus-equivariant cohomology of Nakajima varieties via these subvarieties.
• To provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices in classical groups and their symmetric spaces, particularly in type A.
• To uncover new symmetries and reductions—rectangular symmetry and refined column/row removal—within the geometry of symmetric spaces.

Proposed method

• Define a symplectic involution $\sigma$ as a composition of a diagram automorphism, a reflection functor, and a transpose induced by a bilinear form on the quiver data.
• Construct $\sigma$-quiver varieties as fixed-point subvarieties $\mathfrak{P}_\zeta(\mathbf{v}, \mathbf{w})^{\sigma}$ inside Nakajima quiver varieties $\mathfrak{P}_\zeta(\mathbf{v}, \mathbf{w})$.
• Use the $\hat{\sigma}$-version of the involution to define $\hat{\sigma}$-quiver varieties and relate them to nilpotent Slodowy slices in the symmetric space $\mathfrak{p}$ via a non-degenerate bilinear form $\{-|\cdot\}$ on $\widetilde{V}_i$.
• Establish commutative diagrams identifying $\sigma$-quiver varieties with $\mathcal{S}^{\hat{\sigma}}_{\mu', \lambda}$, the $\hat{\sigma}$-fixed part of a partial Springer resolution.
• Prove that the algebra of $\mathrm{G}^{\tau}_{\mathbf{v}}$-invariant regular functions on $\mathbf{M}(\mathbf{v}, \mathbf{w})^{\hat{\tau}}$ is generated by trace and character functions $\mathrm{tr}_{h_1,\dots,h_s}(-)$ and $\chi_{h_1,\dots,h_s}(-)$.
• Use the Kostant-Sekiguchi-Vergne correspondence to relate the geometry of $\sigma$-quiver varieties to nilpotent orbits in real groups.

Experimental results

Research questions

• RQ1 How can fixed-point subvarieties of Nakajima quiver varieties under symplectic and anti-symplectic involutions serve as a geometric model for symmetric pairs?
• RQ2 What is the relationship between the cohomology of Steinberg-type varieties in $\sigma$-quiver varieties and the universal enveloping algebra of $\mathfrak{k}$?
• RQ3 Can $\sigma$-quiver varieties realize partial Springer resolutions of nilpotent Slodowy slices in classical groups and symmetric spaces?
• RQ4 What new symmetries or reductions emerge in the geometry of $\sigma$-quiver varieties, particularly in type A?
• RQ5 How do the $\hat{\sigma}$-fixed subvarieties relate to the geometry of nilpotent orbits in real forms of $\mathfrak{g}$?

Key findings

• Type A $\sigma$-quiver varieties are isomorphic to partial Springer resolutions of nilpotent Slodowy slices for classical groups, generalizing the Nakajima-Maffei theorem.
• A rectangular symmetry is established: if partitions fit into a rectangle, the associated $\sigma$-quiver varieties are isomorphic, refining previous results.
• The column/row removal reduction is refined in the symmetric space setting, extending results of Kraft and Procesi.
• The $\hat{\sigma}$-fixed subvarieties $\widetilde{\mathcal{S}}^{\hat{\sigma}}_{\mu', \lambda}$ are identified with $\sigma$-quiver varieties via a commutative diagram involving $\pi^{\hat{\sigma}}$ and $\hat{\Pi}$.
• The algebra of $\mathrm{G}^{\tau}_{\mathbf{v}}$-invariant functions on $\mathbf{M}(\mathbf{v}, \mathbf{w})^{\hat{\tau}}$ is generated by trace and character functions along paths.
• Via the Kostant-Sekiguchi correspondence, the geometry of $\sigma$-quiver varieties diffeomorphically corresponds to nilpotent Slodowy slices in real groups.