Minimal Projective Orbits of Semi-simple Lie Groups

2023/02/23 by Henrik Winther, Winther, Henrik
Mathematics · #22E45 #22E46 #57S20 (Primary) 53C10 (Secondary) #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2302.12138

openalex publication_date 2023/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a Lie group G with representation ρ on a real simple G-module \mathbbV. We will call the orbits of the induced action of ρ on the projectivization P\mathbbV the projective orbits, and projective orbits of lowest possible dimension will be called minimal. We show that when G is semi-simple and non-compact, there exists a compact subgroup K⊂ G such that the minimal orbits of G are in bijection with the minimal K-orbits on a K-invariant proper subspace \mathbbW⊂ \mathbbV. In the case that G is split-real, K is the trivial subgroup and there is a unique closed projective orbit, which is moreover of minimal dimension.

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