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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.

53C35 Differential geometry of symmetric spaces
32M10 Homogeneous complex manifolds
22E10 General properties and structure of complex Lie groups