Decomposition theorems for triple spaces (Q2256255)

Let \(G=G_0\times G_o \times G_0\) and \(H=\mathrm{diag}(G_0)\). The corresponding homogeneous space \(G/H\) is called a triple space. In this paper the authors are interested in the triple spaces with \(G_0=\mathrm{SL}(2, K)\), \(K=\mathbb R, \mathbb C\), or \(G_0=\mathrm{SO}_e(n,1)\), \(n=2,3, \hdots\). They show that these spaces admit a polar decomposition \(KAH\) where \(K\) is a maximal compact subgroup, \(A\subset G\) is abelian. They determine for which maximal split abelian subgroups \(A\) the decomposition is valid. It is concluded that there exist maximal split abelian subgroups \(A\) for which \(G= KAH\) and for which \(PH\) is open for all minimal parabolic subgroups \(P\) with \(A\subset P\).