Spherical nilpotent orbits and the Kostant-Sekiguchi correspondence

Abstract:

Let $G$ be a connected, linear semisimple Lie group with Lie algebra $\mathfrak g$, and let ${K_{{}_{\mathbf C}}}~\rightarrow ~{\operatorname {Aut} (\mathfrak p_{{}_{\mathbf C}})}$ be the complexified isotropy representation at the identity coset of the corresponding symmetric space. The Kostant-Sekiguchi correspondence is a bijection between the nilpotent $K_{{}_{\mathbf C}}$-orbits in $\mathfrak p_{{}_{\mathbf C}}$ and the nilpotent $G$-orbits in $\mathfrak g$. We show that this correspondence associates each spherical nilpotent $K_{{}_{\mathbf C}}$-orbit to a nilpotent $G$-orbit that is multiplicity free as a Hamiltonian $K$-space. The converse also holds.