Cycle Spaces of Infinite Dimensional Flag Domains

2015/09/10 by Joseph A. Wolf, Wolf, Joseph A.
Mathematics · #Advanced Algebra and Geometry #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1509.03294

openalex publication_date 2015/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a complex simple direct limit group, specifically SL(∞;ℂ), SO(∞;ℂ) or Sp(∞;ℂ). Let F be a (generalized) flag in ℂ^∞. If G is SO(∞;ℂ) or Sp(∞;ℂ) we suppose further that F is isotropic. Let Z denote the corresponding flag manifold; thus Z = G/Q where Q is a parabolic subgroup of G. In a recent paper with Ignatyev and Penkov, we studied real forms G₀ of G and properties of their orbits on Z. Here we concentrate on open G₀--orbits D ⊂ Z. When G₀ is of hermitian type we work out the complete G₀--orbit structure of flag manifolds dual to the bounded symmetric domain for G₀. Then we develop the structure of the corresponding cycle spaces M_D. Finally we study the real and quaternionic analogs of these theories. All this extends an large body of results from the finite dimensional cases on the structure of hermitian symmetric spaces and related cycle spaces.

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