Double flag varieties for a symmetric pair and finiteness of orbits

Let G be a reductive algebraic group over the complex number filed, and K = G^θ be the fixed points of an involutive automorphism θ of G so that (G,K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal Kaction. The double flag variety G/P × K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P¹ × G/P² × G/P3 is a special case of our double flag varieties, and there are many interesting works on the triple flag varieties. In this paper, we study double flag varieties G/P × K/Q of finite type. We give efficient criterion under which the double flag variety is of finite type. The finiteness of orbits is strongly related to spherical actions of G or K. For example, we show a partial flag variety G/P is K-spherical if a product of partial flag varieties G/P ×G/θ(P) is G-spherical. We also give many examples of the double flag varieties of finite type, and for type AIII, we give a classification when P = B is a Borel subgroup of G.