Journal of Lie Theory

no. 3
An Asymptotic Result on the A-Component in the Iwasawa Decomposition
Journal of Lie Theory, Volume 17 (2007) no. 3, pp. 469-479
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 $G$.
DOI: 10.5802/jolt.456
Classification: 22E46, 22E30
Keywords: Iwasawa decomposition, complete multiplicative Jordan decomposition, Bruhat decomposition, a-component 
@article{JOLT_2007_17_3_a1,
     author = {H. Huang and T.-Y. 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},
     doi = {10.5802/jolt.456},
     zbl = {1163.22006},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.456/}
}
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%A T.-Y. Tam
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H. Huang; T.-Y. Tam. An Asymptotic Result on the A-Component in the Iwasawa Decomposition. Journal of Lie Theory, Volume 17 (2007) no. 3, pp. 469-479. doi: 10.5802/jolt.456

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