Some remarks on real minimal nilpotent orbits and symmetric pairs

Abstract

For a non-compact simple Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, we denote by $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}}$ the unique complex nilpotent orbit in $\mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$ containing all minimal real nilpotent orbits in $\mathfrak{g}$. In this paper, we give a complete classification of symmetric pairs $(\mathfrak{g},\mathfrak{h})$ such that $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}} \cap \mathfrak{g}^d = \emptyset$, where $\mathfrak{g}^d$ denotes the dual Lie algebra of $(\mathfrak{g},\mathfrak{h})$. Furthermore, for symmetric pairs $(G,H)$ with real simple Lie group $G$, we apply our classification to theorems given by T. Kobayashi [J. Lie Theory (2023)], and study bounded multiplicity properties of restrictions on $H$ of infinite-dimensional irreducible $G$-representations with minimum Gelfand--Kirillov dimension.