Some remarks on real minimal nilpotent orbits and symmetric pairs

For a non-compact simple Lie algebra g over R, we denote by O^C, ₆ the unique complex nilpotent orbit in g R C containing all minimal real nilpotent orbits in g. In this paper, we give a complete classification of symmetric pairs (g, h) such that O^C, ₆ gᵈ =, where gᵈ denotes the dual Lie algebra of (g, 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.