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 für 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).