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On the similarity between the Iwasawa projection and the diagonal part. (English) Zbl 0564.22007
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.
The main results of this paper is the following theorem. There is a real analytic map \(\psi\) : \(s\to K\) such that \((i)\quad \phi_ X: k\to k\cdot \psi (Ad k^{-1}(X))\) is a diffeomorphism from K onto K, for each \(X\in s\), \((ii)\quad \gamma (Ad \psi (X)^{-1}(X))=\pi (X)\) for all \(X\in s\). - This means that the Iwasawa projection can be turned into the orthogonal projection \(\pi\) by the action \(Ad \psi (X)^{-1}\in Ad K,\) and the element \(\psi\) (X) of K depends analytically on \(X\in s\). Some results obtained by J. J. Duistermaat, J. A. C. Kolk and V. S. Varadarajan [Compos. Math. 49, 309-398 (1983; Zbl 0524.43008)] are used in the argument of the theorem of this paper. For \(G=SL(2,{\mathbb{R}})\), an explicit consideration is also given.
Reviewer: Ch.Cheng

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