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 $G$.
1
Dept. of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, U.S.A.
Huajun Huang; Tin-Yau Tam. An Asymptotic Result on the A-Component in the Iwasawa Decomposition. Journal of Lie Theory, Tome 17 (2007) no. 3, pp. 469-479. http://geodesic.mathdoc.fr/item/JOLT_2007_17_3_a1/
@article{JOLT_2007_17_3_a1,
author = {Huajun Huang and Tin-Yau Tam},
title = {An {Asymptotic} {Result} on the {A-Component} in the {Iwasawa} {Decomposition}},
journal = {Journal of Lie Theory},
pages = {469--479},
year = {2007},
volume = {17},
number = {3},
zbl = {1163.22006},
url = {http://geodesic.mathdoc.fr/item/JOLT_2007_17_3_a1/}
}
TY - JOUR
AU - Huajun Huang
AU - Tin-Yau Tam
TI - An Asymptotic Result on the A-Component in the Iwasawa Decomposition
JO - Journal of Lie Theory
PY - 2007
SP - 469
EP - 479
VL - 17
IS - 3
UR - http://geodesic.mathdoc.fr/item/JOLT_2007_17_3_a1/
ID - JOLT_2007_17_3_a1
ER -
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%A Tin-Yau Tam
%T An Asymptotic Result on the A-Component in the Iwasawa Decomposition
%J Journal of Lie Theory
%D 2007
%P 469-479
%V 17
%N 3
%U http://geodesic.mathdoc.fr/item/JOLT_2007_17_3_a1/
%F JOLT_2007_17_3_a1
