Let G be a complex simple direct limit group, specifically
SL(β;C), SO(β;C) or Sp(β;C).
Let F be a (generalized) flag in Cβ. If G is
SO(β;C) or Sp(β;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 G0β of G and
properties of their orbits on Z. Here we concentrate on open
G0β--orbits DβZ. When G0β is of hermitian type we work
out the complete G0β--orbit structure of flag manifolds dual to the bounded
symmetric domain for G0β. Then we develop the structure of the corresponding
cycle spaces MDβ. 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.Comment: This revision improves the exposition and corrects a number of typos.
Earlier revisions had clarified the ordering of subspaces in a flag relative
to a given ordered basis of the ambient Cβ as well as the
product structure of the base cycles for flag domains of Sp(β;R) and
SOβ(β). These revisions had no effect on the results for the
structure of the cycle space