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The Affine Closure of T^*(SL_n/U)

Boming Jia

TL;DR

The paper proves that the affine closure $\overline{T^*(G/U)}$ has symplectic singularities for $G=\mathrm{SL}_n$, using a quiver-variety realization and Hamiltonian reduction to model $\overline{T^*(\mathrm{SL}_n/U)}$ as $N\sslash H$ with codimension-4 singularities. In the pivotal case $n=3$, it constructs an explicit $\mathrm{SL}_2$-equivariant symplectic isomorphism $F$ identifying $\overline{T^*(\mathrm{SL}_3/U)}$ with the closure of the minimal nilpotent orbit $\overline{\mathcal{O}}_{\min} \subset \mathfrak{so}_8$, and interprets the Weyl-group action (Gelfand–Graev action) as the triality action on $\mathfrak{so}_8$ restricted to this orbit. The work blends quiver variety techniques, Hamiltonian reduction, and Lie-theoretic symmetries to connect affine closures, minimal orbits, and $D_4$-type triality, providing concrete models and symmetry identifications. Overall, it illuminates the geometric structure and symmetries of $\overline{T^*(G/U)}$, offering tools to study deformations and quantizations in this setting.

Abstract

We show that the affine closure of T^*(SL_n/U) has symplectic singularities, in the sense of Beauville. In the special case n=3, we show that the affine closure of T^*(SL_3/U) is isomorphic to the closure of the minimal nilpotent adjoint orbit in so(8,C). Moreover, the quasi-classical Gelfand-Graev action of the Weyl group W on the affine closure of T^*(SL_3/U) can be identified with the restriction to the closure of the minimal nilpotent adjoint orbit of the triality action on so(8,C).

The Affine Closure of T^*(SL_n/U)

TL;DR

The paper proves that the affine closure has symplectic singularities for , using a quiver-variety realization and Hamiltonian reduction to model as with codimension-4 singularities. In the pivotal case , it constructs an explicit -equivariant symplectic isomorphism identifying with the closure of the minimal nilpotent orbit , and interprets the Weyl-group action (Gelfand–Graev action) as the triality action on restricted to this orbit. The work blends quiver variety techniques, Hamiltonian reduction, and Lie-theoretic symmetries to connect affine closures, minimal orbits, and -type triality, providing concrete models and symmetry identifications. Overall, it illuminates the geometric structure and symmetries of , offering tools to study deformations and quantizations in this setting.

Abstract

We show that the affine closure of T^*(SL_n/U) has symplectic singularities, in the sense of Beauville. In the special case n=3, we show that the affine closure of T^*(SL_3/U) is isomorphic to the closure of the minimal nilpotent adjoint orbit in so(8,C). Moreover, the quasi-classical Gelfand-Graev action of the Weyl group W on the affine closure of T^*(SL_3/U) can be identified with the restriction to the closure of the minimal nilpotent adjoint orbit of the triality action on so(8,C).
Paper Structure (6 sections, 16 theorems, 104 equations)

This paper contains 6 sections, 16 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The Affine Closure $\overline{T^*(\mathrm{SL}_{n}/U)}$ Has Symplectic Singularities.
  3. Isomorphism between $\overline{T^*(\mathrm{SL}_3/U)}$ and $\overline{\mathcal{O}}_\textrm{min}\subset\mathfrak{so}_8$
  4. The Triality Action on $\mathfrak{so}_8$
  5. The Gelfand-Graev Action
  6. Symplectic form on the smooth locus

Key Result

Theorem 2.1

The affine closure $\overline{T^*(\mathrm{SL}_n/U)}$ is isomorphic to the categorical quotient $N\sslash H$ as affine varieties.

Theorems & Definitions (40)

  • Conjecture 1.1
  • Remark 2.1
  • Theorem 2.1: Theorem 7.18 in DKS
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 30 more