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Abstract: Let G be a reductive algebraic group over the complex number filed, and K =G^{\theta} be the fixed points of an involutive automorphism \theta of G sothat G, K is a symmetric pair. We take parabolic subgroups P and Q of G and Krespectively and consider a product of partial flag varieties G-P and K-Q withdiagonal K-action. The double flag variety G-P \times K-Q thus obtained is saidto be of finite type if there are finitely many K-orbits on it. A triple flagvariety G-P^1 \times G-P^2 \times G-P^3 is a special case of our double flagvarieties, and there are many interesting works on the triple flag varieties.In this paper, we study double flag varieties G-P \times K-Q of finite type. Wegive 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 ofpartial flag varieties G-P \times G-\thetaP is G-spherical. We also give manyexamples of the double flag varieties of finite type, and for type AIII, wegive a classification when P = B is a Borel subgroup of G.



Autor: Kyo Nishiyama, Hiroyuki Ochiai

Fuente: https://arxiv.org/







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