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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.

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Some Remarks on Real Minimal Nilpotent Orbits and Symmetric Pairs

  • 会议论文

  • 发表日期：

    2026年

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  • Takayuki Okuda

摘要

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.