Orbits and invariants associated with a pair of spherical varieties Some examples

Let H and K be spherical subgroups of a reductive complex group G. In many cases, detailed knowledge of the double coset space H\G/K is of fundamental importance in group theory and representation theory. If H or K is parabolic, then H\G/K is finite, and we recall the classification of the double cosets in several important cases. If H = K is a symmetric subgroup of G, then the double coset space K\G/K (and the corresponding invariant theoretic quotient) are no longer finite, but several nice properties hold, including an analogue of the Chevalley restriction theorem. In (Helminck and Schwarz, 2001) these properties were generalized to the case where H and K are fixed point groups of commuting involutions. We recall the main results of (Helminck and Schwarz, 2001). We also give examples to show the difficulty in extending these results if we allow H = K to be a reductive spherical (non-symmetric) subgroup or if we have H symmetric and K spherical reductive.