On orbits in double flag varieties for symmetric pairs

Let $G$ be a connected, simply connected semisimple algebraic group over the complex number field, and let $K$ be the fixed point subgroup of an involutive automorphism of $G$ so that $(G, K)$ is a symmetric pair. We take parabolic subgroups $P$ of $G$ and $Q$ of $K$ respectively and consider the product of partial flag varieties $G/P$ and $K/Q$ with diagonal $K$-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $K$-orbits on it. In this paper, we give a parametrization of $K$-orbits on $G/P \times K/Q$ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $P \subset G$ or $Q \subset K$ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $K$-spherical flag varieties $G/P$ and $G$-spherical homogeneous spaces $G/Q$.

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