
Journal of Lie Theory 17 (2007), No. 3, 469479 Copyright Heldermann Verlag 2007 An Asymptotic Result on the AComponent in the Iwasawa Decomposition Huajun Huang Dept. of Mathematics and Statistics, Auburn University, Auburn, AL 368495310, U.S.A. huanghu@auburn.edu TinYau Tam Dept. of Mathematics and Statistics, Auburn University, Auburn, AL 368495310, U.S.A. tamtiny@auburn.edu [Abstractpdf] Let $G$ be a connected noncompact semisimple Lie group. For each $v', v, g\in G$, we prove that $$\lim_{t\to \infty} [a(v'g^tv)]^{1/t} = s^{1} \cdot b(g),$$ where $a(g)$ denotes the $a$component in the Iwasawa decomposition of $g = kan$ and $b(g)\in A_+$ denotes the unique element that is conjugate to the hyperbolic component $h$ in the complete multiplicative Jordan decomposition of $g = ehu$. The element $s$ in the Weyl group of $(G,A)$ is determined by $yv\in G$ (not unique in general) in such a way that $yv\in N^m_sMAN$, where $yhy^{1}=b(g)$ and $G = \cup_{s\in W} N^ m_sMAN$ is the Bruhat decomposition of $G$. Keywords: Iwasawa decomposition, complete multiplicative Jordan decomposition, Bruhat decomposition, acomponent. MSC: 22E46; 22E30 [ Fulltextpdf (192 KB)] for subscribers only. 