Orbits and invariants associated with a pair of spherical varieties: some examples. (Q1862244)

Let \(H\) and \(K\) be spherical subgroups of a complex reductive algebraic group \(G\). If \(H\) and \(K\) are parabolic, the authors recall the classification of the double \(H\)-\(K\)-cosets of \(G\) (i.e., the elements of \(H\backslash G/K\)). If \(H\) and \(K\) are fixed point groups of commuting involutions of \(G\), they recall the main results of \textit{A. G. Helminck} and \textit{G. W. Schwarz} [Duke Math. J. 106, No. 2, 237--279 (2001; Zbl 1015.20031)]. It are also given examples to show the difficulty in extending these results if one allows \(H=K\) to be a reductive spherical (nonsymmetric) subgroup or if \(H\) is symmetric and \(K\) is spherical.