The affine closure of cotangent bundles of horospherical spaces

TL;DR

This paper proves that the affine closure of cotangent bundles of certain homogeneous spaces, specifically horospherical spaces, is a symplectic variety, extending previous results and providing a new geometric proof.

Contribution

It introduces a geometric approach to show that the affine closure of cotangent bundles of horospherical spaces is symplectic, generalizing prior work on specific cases.

Findings

Affine closure of cotangent bundles of horospherical spaces is symplectic.

The result applies to all quasi-affine homogeneous spaces with horospherical stabilizers.

In particular, affine closures of cotangent bundles of G/[P,P] are symplectic for any parabolic subgroup P.

Abstract

For a smooth quasi-affine variety $X$, the affine closure $\overline{T^*X} := \text{Spec}(\mathbb{K}[T^*X])$ contains $T^*X$ as an open subset, and its smooth locus carries a symplectic structure. A natural question is whether $\overline{T^*X}$ itself is a symplectic variety. A notable example is the conjecture of Ginzburg and Kazhdan, which predicts that $\overline{T^*(G/U)}$ is symplectic for a maximal unipotent subgroup $U$ in a reductive linear algebraic group $G$. This conjecture was recently proved by Gannon using representation-theoretic methods. In this paper, we provide a new geometric approach to this conjecture. Our method allows us to prove a more general result: $\overline{T^*(G/H)}$ is symplectic for any horospherical subgroup $H$ in $G$ such that $G/H$ is quasi-affine. In particular, this implies that the affine closure $\overline{T^*(G/[P,P])}$ is a symplectic variety for any parabolic subgroup $P$ in $G$.