2012/08/10 by Xuhua He, Kyo Nishiyama, He, Xuhua +5
Mathematics · #14M15 (Primary) 53C35 #14M17 (Secondary) #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:14M15 #msc:14M17 #msc:53C35
paper · pdf · doi:10.48550/arxiv.1208.2084
47 pages, 3 tables; add all the details of the classification
arxiv created 2013/07/27 · arxiv updated 2013/07/30
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 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 .