Cycle spaces of infinite dimensional flag domains

Abstract

Let G be a complex simple direct limit group, specifically $$SL(\infty ;\mathbb {C})$$ , $$SO(\infty ;\mathbb {C})$$ or $$Sp(\infty ;\mathbb {C})$$ . Let $$\mathcal {F}$$ be a (generalized) flag in $$\mathbb {C}^\infty $$ . If G is $$SO(\infty ;\mathbb {C})$$ or $$Sp(\infty ;\mathbb {C})$$ we suppose further that $$\mathcal {F}$$ is isotropic. Let $$\mathcal {Z}$$ denote the corresponding flag manifold; thus $$\mathcal {Z}= G/Q$$ where Q is a parabolic subgroup of G. In a recent paper (Penkov and Wolf in Real group orbits on flag ind-varieties of $$SL_{\infty } ({\mathbb {C}})$$ , to appear in Proceedings in Mathematics and Statistics) we studied real forms $$G_0$$ of G and properties of their orbits on $$\mathcal {Z}$$ . Here we concentrate on open $$G_0$$ -orbits $$D \subset \mathcal {Z}$$ . When $$G_0$$ is of hermitian type we work out the complete $$G_0$$ -orbit structure of flag manifolds dual to the bounded symmetric domain for $$G_0$$ . Then we develop the structure of the corresponding cycle spaces $$\mathcal {M}_D$$ . Finally we study the real and quaternionic analogs of these theories. All this extends results from the finite-dimensional cases on the structure of hermitian symmetric spaces and cycle spaces (in chronological order: Wolf in Bull Am Math Soc 75:1121–1237, 1969; Wolf et al. in Ann Math 105:397–448, 1977; Wolf in Ann Math 136:541–555, 1992; Wolf in Compact subvarieties in flag domains, 1994; Wolf and Zierau in Math Ann 316:529–545, 2000; Huckleberry et al. in Journal fur die reine und angewandte Mathematik 2001:171–208, 2001; Huckleberry and Wolf in Cycle spaces of real forms of $$SL_n(C)$$ , Springer, New York, pp 111–133, 2002; Wolf and Zierau in J Lie Theory 13:189–191, 2003; Huckleberry and Wolf in Ann Scuola Norm Sup Pisa Cl Sci (5) 9:573-580, 2010).