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

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.