Some remarks on real minimal nilpotent orbits and symmetric pairs

For a non-compact simple Lie algebra g over R, we denote by OCmin,g 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 OCmin,g ∩ 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.