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A generalized Cartan decomposition for the double coset space $$\mathrm{SU}(2n+1)\backslash \mathrm{SL}(2n+1,\mathbb C)/\mathrm{Sp}(n,\mathbb C)$$. (English) Zbl 1251.53031
Let $$G$$ be a non–compact reductive Lie group, $$H$$ a closed subgroup of $$G$$, and $$K$$ a maximal compact subgroup of $$G$$. For symmetric spaces $$G\big/H$$ it is known that there is an analogue of the Cartan decomposition $$G = KAH$$, where $$A$$ is a non–compact abelian subgroup of $$G$$. For general non–symmetric spaces $$G\big/H$$ there does not always exist an abelian subgroup $$A$$ such that the multiplication map $$K\times A\times H \to G$$ is surjective (cf. [T. Kobayashi, J. Reine Angew. Math. 490, 37–54 (1997; Zbl 0881.22013)]). However, if $$K$$ acts on $$G\big/H$$ in a visible fashion in the sense of T. Kobayashi [Acta Appl. Math. 81, No. 1-3, 129–146 (2004; Zbl 1050.22018)], then a nice decomposition $$G=KAH$$ for some nice subgroup (or subset) $$A$$ may be expected even for non–symmetric spaces $$G\big/H$$.
In the present paper, the author gives a generalization of the Cartan decomposition for the non–symmetric space $$G_{\mathbb C}\big/H_{\mathbb C} = SL(2n+1,\mathbb C)\big/ Sp(n,\mathbb C)$$. Taking a maximal compact subgroup $$G_u = SU(2n+1)$$ of $$G_{\mathbb C}$$, the author proves that there exists a $$2n$$–dimensional ’slice’ $$A$$ in $$G_{\mathbb C}$$ such that $$G_{\mathbb C} = G_uAH_{\mathbb C}$$. As a corollary the author proves that the action of $$SU(2n+1)$$ on $$G_{\mathbb C}\big/H_{\mathbb C}$$ is strongly visible.

##### MSC:
 53C35 Differential geometry of symmetric spaces 32M10 Homogeneous complex manifolds 22E10 General properties and structure of complex Lie groups
##### Keywords:
Cartan decomposition; visible action; symmetric space