A generalized Cartan decomposition for the double coset space \(\mathrm{SU}(2n+1)\backslash \mathrm{SL}(2n+1,\mathbb C)/\mathrm{Sp}(n,\mathbb C)\) (Q2902432)

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. [\textit{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 \textit{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 \(G\big/H\).NEWLINENEWLINEIn 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.