On Orbits in Double Flag Varieties for Symmetric Pairs

Xuhua He, Hiroyuki Ochiai, Kyo Nishiyama, Yoshiki Oshima

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22 Citations (Scopus)


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 languageEnglish
Pages (from-to)1091-1136
Number of pages46
JournalTransformation Groups
Issue number4
Publication statusPublished - Dec 2013

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Geometry and Topology


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