Growth estimates for orbits of self-adjoint groups

Abstract

Let $$G$$ denote a closed, connected, self-adjoint, noncompact subgroup of $$GL(n,\mathbb R )$$ , and let $$d_{R}$$ and $$d_{L}$$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $$M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$$ . Let $$v$$ be a nonzero vector of $$\mathbb R ^{n}$$ such that the orbit $$G(v)$$ is unbounded in $$\mathbb R ^{n}$$ . Then the function $$g \rightarrow d_{R}(g, G_{v})$$ is unbounded, where $$G_{v} = \{g \in G : g(v) = v \}$$ , and we obtain algebraically defined upper and lower bounds $$\lambda ^{+}(v)$$ and $$\lambda ^{-}(v)$$ for the asymptotic behavior of the function $$\frac{log|g(v)|}{d_{R}(g, G_{v})}$$ as $$d_{R}(g, G_{v}) \rightarrow \infty $$ . The upper bound $$\lambda ^{+}(v)$$ is at most 1. The orbit $$G(v)$$ is closed in $$\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$$ is positive for some w $$\in G(v)$$ . If $$G_{v}$$ is compact, then $$g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$$ is uniformly bounded in $$G$$ , and the exponents $$\lambda ^{+}(v)$$ and $$\lambda ^{-}(v)$$ are sharp upper and lower asymptotic bounds for the functions $$\frac{log|g(v)|}{d_{R}(g,I)}$$ and $$\frac{log|g(v)|}{d_{L}(g,I)}$$ as $$d_{R}(g,I) \rightarrow \infty $$ or as $$d_{L}(g,I) \rightarrow \infty $$ . However, we show by example that if $$G_{v}$$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $$\frac{log|g(v)|}{d_{L}(g, G_{v})}$$ as $$d_{L}(g, G_{v}) \rightarrow \infty $$ . The results apply to representations of noncompact semisimple Lie groups $$G$$ on finite dimensional real vector spaces. We compute $$\lambda ^{+}$$ and $$\lambda ^{-}$$ for the irreducible, real representations of $$SL(2,\mathbb R )$$ , and we show that if the dimension of the $$SL(2,\mathbb R )$$ -module $$V$$ is odd, then $$\lambda ^{+} = \lambda ^{-}$$ on a nonempty open subset of $$V$$ . We show that the function $$\lambda ^{-}$$ is $$K$$ -invariant, where $$K = O(n,\mathbb R ) \cap G$$ . We do not know if $$\lambda ^{-}$$ is $$G$$ -invariant.