Growth Estimates for Orbits of Self Adjoint Groups

TL;DR

This paper establishes algebraic bounds on the growth rate of group orbits in real vector spaces, linking orbit closure properties to growth behavior, with applications to semisimple Lie group representations.

Contribution

It provides new algebraic bounds for orbit growth rates in self-adjoint group actions, characterizing when orbits are closed based on growth positivity.

Findings

Bounds are sharp when stabilizer groups are compact.

Positive lower bounds imply orbit closure.

Results apply to semisimple Lie group representations.

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 R^{n} 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 R^{n}. The results apply to representations of noncompact semisimple Lie groups G on finite dimensional real vector spaces.