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 ${\overline{T^{\ast }{\mathbb{O}}_n}}^{\mathrm{aff}}$ is isomorphic to a ${\mathbb{C}}^{\ast }$-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 ${\overline{T^{\ast }{\mathbb{O}}_n}}^{\mathrm{aff}}$ has no symplectic resolution.

arXiv comment26 pages, minor revision. Comments are welcome