### Abstract

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 double flag variety for a symmetric pair. It is said to be 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 × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ 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.

Original language | English |
---|---|

Pages (from-to) | 1091-1136 |

Number of pages | 46 |

Journal | Transformation Groups |

Volume | 18 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1 2013 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology

## Fingerprint Dive into the research topics of 'On Orbits in Double Flag Varieties for Symmetric Pairs'. Together they form a unique fingerprint.

## Cite this

*Transformation Groups*,

*18*(4), 1091-1136. https://doi.org/10.1007/s00031-013-9243-8