Some remarks on real minimal nilpotent orbits and symmetric pairs

2024/06/30 by Okuda, Takayuki
#32M15 #53C15 #53C35 #57S30 #FOS: Mathematics #Primary 22E46 #Representation Theory (math.RT) #Secondary 22E45

paper · doi:10.48550/arxiv.2407.00675

Abstract

For a non-compact simple Lie algebra \mathfrakg over ℝ, we denote by O^ℂ_min,\mathfrakg the unique complex nilpotent orbit in \mathfrakg ⊗_ℝ ℂ containing all minimal real nilpotent orbits in \mathfrakg. In this paper, we give a complete classification of symmetric pairs (\mathfrakg,\mathfrakh) such that O^ℂ_min,\mathfrakg ∩ \mathfrakg^d = ∅, where \mathfrakg^d denotes the dual Lie algebra of (\mathfrakg,\mathfrakh). 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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