The affine closure of cotangent bundles of horospherical spaces
Baohua Fu, Jie Liu
TL;DR
The paper develops a geometric framework for the affine closures $\overline{T^*X}$ of cotangent bundles of quasi-affine horospherical spaces $X=G/H$, proving that $S(G/H)$ is finitely generated and $\overline{T^*(G/H)}$ is a symplectic variety when $H$ is horospherical and $G/H$ is quasi-affine. The authors deploy Cox rings and torus invariants to relate $S(G/H)$ to the Cox ring of a weak Fano projective bundle $\mathbb{P}E^*$, establishing rational singularities and (Q-)factoriality under conditions on $\mathrm{Cl}(X)$; these ingredients yield corollaries for parabolic basic spaces $G/[P,P]$ as well as corank-one degenerations. They also analyze cones of highest weight vectors (HV cones) attached to IHSSs, showing that for certain IHSS data the affine closure $\overline{T^*X}$ coincides with the minimal nilpotent orbit closure $\overline{\mathcal{O}}_{\min}$ in $\mathfrak{g}$, thereby identifying broad classes of symplectic closures. The results connect finite generation, Mori dream space structure, and Fano-type geometry to symplectic singularities arising from cotangent bundles, offering a unifying perspective for Ginzburg–Kazhdan type conjectures beyond $G/U$. Overall, the work provides a robust, geometry-driven route to establishing symplecticity and regularity properties of affine closures in the horospherical setting and beyond.
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 $G$.