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Journal of Lie Theory 21 (2011), No. 1, 079--099
Copyright Heldermann Verlag 2011

Double Flag Varieties for a Symmetric Pair and Finiteness of Orbits

Kyo Nishiyama
Dept. of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1 Chuo, Sagamihara 252-5258, Japan

Hiroyuki Ochiai
Faculty of Mathematics, Kyushu University, 744 Motooka Nishi-ku, Fukuoka 819-0395, Japan


Let $G$ be a reductive algebraic group over the complex number field, and $K = G^{\theta}$ be the fixed points of an involutive automorphism $\theta$ of $G$ so that $(G, K)$ is a symmetric pair. \endgraf 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 $K$-action. The double flag variety $G/P \times K/Q$ thus obtained is said to be {\it of finite type} if there are finitely many $K$-orbits on it. A triple flag variety $G/P^1 \times G/P^2 \times G/P^3$ is a special case of our double flag varieties, and there are many interesting works on the triple flag varieties. \endgraf In this paper, we study double flag varieties $G/P \times K/Q$ 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 $G/P \times G/\theta(P)$ 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 $G$.

Keywords: Symmetric pair, flag variety, spherical action.

MSC: 14M15; 53C35, 14M17

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