Growth Estimates for Orbits of Self Adjoint Groups

2010/12/13 by Patrick Eberlein, Eberlein, Patrick
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1012.2898

openalex publication_date 2010/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G denote a closed, connected, self adjoint, noncompact subgroup of GL(n,R), and let d_R denote the canonical right invariant Riemannian metric on G. For v in Rⁿ let G_v = g in G : g(v) = v. We obtain algebraically defined upper and lower bounds for the asymptotic growth rate of g --> log |g(v)| / d_R(g,G_v), and these bounds are sharp if G_v is compact. If the lower bound is positive, then the orbit G(v) is closed in Rⁿ. The results apply to representations of noncompact semisimple Lie groups G on finite dimensional real vector spaces.

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