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

We establish a novel connection between the minimal nilpotent orbit ${\mathbb{O}}_n$ in ${\mathfrak{s}\mathfrak{l}}_n$ and the minimal nilpotent orbit closure ${\bar{\mathbf{O}}}_n$ in ${\mathfrak{s}\mathfrak{o}}_{2n+2}$, 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 ${\overline{T^{\ast }{\mathbb{O}}_n}}^{\mathrm{a}\mathrm{f}\mathrm{f}}$ is isomorphic to a ${\mathbb{C}}^{\ast }$-Hamiltonian reduction of ${\bar{\mathbf{O}}}_n$. 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 ${\overline{T^{\ast }{\mathbb{O}}_n}}^{\mathrm{a}\mathrm{f}\mathrm{f}}$ has no symplectic resolution.