# zbMATH — the first resource for mathematics

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:
 2.2e+16 General properties and structure of real Lie groups
Full Text:
##### References:
 [1] J.-L. Clerc , On the asymptotic behaviour of generalized Bessel functions , Rend. Circ. Mat. Palermo (2) 1981 , Supp. No. 1, pp. 145-147. MR 83c:58078 | Zbl 0512.33011 · Zbl 0512.33011 [2] J.J. Duistermaat , J.A.C. Kolk and V.S. Varadarajan , Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semi-simple Lie groups , Comp. Math. 49 ( 1983 ), 309-398. Numdam | MR 85e:58150 | Zbl 0524.43008 · Zbl 0524.43008 · numdam:CM_1983__49_3_309_0 · eudml:89615 [3] G.J. Heckman , Projections of Orbits and Asymptotic Behaviour of Multiplicities for Compact Lie Groups , Thesis, Rijksuniversiteit Leiden, 1980 . [4] B. Kostant , On convexity, the Weyl group and the Iwasawa decomposition , Ann. Sci. Éc. Norm. Sup. 6 ( 1973 ), 413-455. Numdam | MR 51 #806 | Zbl 0293.22019 · Zbl 0293.22019 · numdam:ASENS_1973_4_6_4_413_0 · eudml:81923 [5] J.N. Mather , Infinitesimal stability implies stability , Ann. of Math. 89 ( 1969 ), 254-291. MR 41 #4582 | Zbl 0177.26002 · Zbl 0177.26002 · doi:10.2307/1970668 [6] J. Moser , On the volume elements on a manifold , Trans. A.M.S. 120 ( 1965 ), 286-294. MR 32 #409 | Zbl 0141.19407 · Zbl 0141.19407 · doi:10.2307/1994022 [7] R.J. Stanton and P.A. Tomas , Expansions for spherical functions on noncompact symmetric spaces , Acta Math. 140 ( 1978 ), 251-276. MR 58 #23365 | Zbl 0411.43014 · Zbl 0411.43014 · doi:10.1007/BF02392309 [8] T. Koornwinder , A new proof of a Paley-Wiener type theorem for the Jacobi transform , Arkiv för Matematik, 13 ( 1975 ), 145-159. MR 51 #11028 | Zbl 0303.42022 · Zbl 0303.42022 · doi:10.1007/BF02386203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.