A connection between minimal nilpotent orbits of types A and D via Hamiltonian reduction

TL;DR

This paper uncovers a new geometric link between minimal nilpotent orbits in types A and D via Hamiltonian reduction, contrasting with previous shared-orbit paradigms.

Contribution

It demonstrates that the affine closure of the cotangent bundle of a type-A nilpotent orbit is isomorphic to a Hamiltonian reduction of a type-D orbit closure, revealing a novel type-A--D connection.

Findings

Affine closure of cotangent bundle is isomorphic to a Hamiltonian reduction.

No symplectic resolution exists for the affine closure.

Provides a classical analogue of a quantum result by Levasseur and Stafford.

Abstract

We establish a novel connection between the minimal nilpotent orbit $\mathbb{O}_n$ in $\mathfrak{sl}_n$ and the minimal nilpotent orbit closure $\overline{\mathbf{O}}_n$ in $\mathfrak{so}_{2n+2}$, which differs from the shared-orbit paradigm of Brylinski and Kostant, where no direct type-A--type-D relation appears. More precisely, we show that the affine closure of the cotangent bundle $\overline{T^*\mathbb{O}_n}^{\mathrm{aff}}$ is isomorphic to a $\mathbb{C}^*$-Hamiltonian reduction of $\overline{\mathbf{O}}_n$. This provides a quasi-classical analogue of a quantum result of Levasseur and Stafford. A detailed study of the geometry of this Hamiltonian reduction reveals that $\overline{T^*\mathbb{O}_n}^{\mathrm{aff}}$ has no symplectic resolution.