Some remarks on real minimal nilpotent orbits and symmetric pairs

TL;DR

This paper classifies symmetric pairs of real simple Lie algebras based on the intersection properties of minimal complex nilpotent orbits with dual Lie algebras, and explores implications for representation theory.

Contribution

It provides a complete classification of symmetric pairs with specific orbit intersection properties and applies this to analyze bounded multiplicity in representation restrictions.

Findings

Classified symmetric pairs with empty intersection of minimal nilpotent orbits

Applied classification to study restriction properties of infinite-dimensional representations

Extended results to bounded multiplicity phenomena in representation theory

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.