Some remarks on real minimal nilpotent orbits and symmetric pairs

For a non-compact simple Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, we denote by ${\mathcal{O}}_{\min ,\mathfrak{g}}^{\mathbb{C}}$ 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 ${\mathcal{O}}_{\min ,\mathfrak{g}}^{\mathbb{C}}\cap {\mathfrak{g}}^d=\mathrm{\varnothing }$, 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.