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Zbl 0564.22007
Duistermaat, J.J.
On the similarity between the Iwasawa projection and the diagonal part. (English)
[J] Mém. Soc. Math. Fr., Nouv. Sér. 15, 129-138 (1984). ISSN 0249-633X

Let G be a real connected semi-simple Lie group with finite center and $G=KAN$ be its Iwasawa decomposition. Let g be the Lie algebra of G and k be the Lie algebra of K. Let s denote the orthogonal complement of k in g with respect to the Killing form, (s always denoted by p), then the Cartan decomposition $G=K\cdot \exp s$ yields that $s\to G\to K\setminus G$ is a diffeomorphism from s onto the (non-compact Riemannian) symmetric space $K\setminus G$. The Iwasawa projection H from G onto the Lie algebra a of A is defined by $x\in K\cdot \exp H(x)\cdot N$, $x\in G$. Let $\gamma$ be the mapping $H\cdot \exp: s\to a$, and let $\pi$ be the orthogonal projection $s\to a$ with respect to the Killing form. \par The main results of this paper is the following theorem. There is a real analytic map $\psi$ : $s\to K$ such that $(i)\quad \phi\sb X: k\to k\cdot \psi (Ad k\sp{-1}(X))$ is a diffeomorphism from K onto K, for each $X\in s$, $(ii)\quad \gamma (Ad \psi (X)\sp{-1}(X))=\pi (X)$ for all $X\in s$. - This means that the Iwasawa projection can be turned into the orthogonal projection $\pi$ by the action $Ad \psi (X)\sp{-1}\in Ad K,$ and the element $\psi$ (X) of K depends analytically on $X\in s$. Some results obtained by {\it J. J. Duistermaat}, {\it J. A. C. Kolk} and {\it V. S. Varadarajan} [Compos. Math. 49, 309-398 (1983; Zbl 0524.43008)] are used in the argument of the theorem of this paper. For $G=SL(2,{\bbfR})$, an explicit consideration is also given.
[Ch.Cheng]

MSC 2000:: *22E15 General properties and structure of real Lie groups

Keywords: real connected semi-simple Lie group; Iwasawa decomposition; Killing form; symmetric space; Iwasawa projection

Citations: Zbl 0524.43008

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