Journal of Lie Theory 17 (2007), No. 3, 469--479
Copyright Heldermann Verlag 2007
An Asymptotic Result on the A-Component in the Iwasawa Decomposition
Huajun Huang
Dept. of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, U.S.A.
huanghu@auburn.edu
Tin-Yau Tam
Dept. of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, U.S.A.
tamtiny@auburn.edu
[Abstract-pdf]
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, a-component.
MSC: 22E46; 22E30
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