Let G be a complex reductive algebraic group, and B a Borel subgroup.
It is well known that a symmetric subgroup K fixed by an involution has
finitely many orbits on the flag variety G/B. This fact is an important
tool of the theory of Harish-Chandra modules of the symmetric pair (G,K)
through the K-equivariant D-modules.
In the talk, we give a variety of examples of products of (partial) flag
varieties which admit finitely many K-orbits. If the pair is (Gtimes
G, diag(G)), a complete classification for the classical groups is
already given by the works of Magyar-Weymann-Zelevinsky. We generalize
their results to get a wider class of such multiple flag varieties for a
general symmetric pair. As a consequence, we also get various
criterions for spherical G- or K-actions.
This talk is based on a joint work with Hiroyuki Ochiai (Kyushu Univeristy).