Description:
We consider the complex ind-group G=SL (∞, C) and its real forms G0=SU(∞,∞), SU(p,∞), SL(∞,R), SL(∞,H). Our main object of study are the G0-orbits on an ind-variety G/P for an arbitrary splitting parabolic ind-subgroup P c G, under the assumption that the subgroups G0cG and PcG are aligned in a natural way. We prove thest the intersection of any G0-orbit on G/P with a finite-dimensional flag variety Gn/Pn from a given exhaustion of G/P via Gn/Pn for n - ∞, is a single (G0 ∩ Gn)-orbit. We also characterize all ind-varieties G/P on which there are finitely many G0-orbits, and provide criteria for the existence of open and close G0-orbits on G/P in the case of infinitely many G0-orbits.