On Homogeneous Spaces for Diagonal Ind-Groups

We study the homogeneous ind-spaces \(\textrm{GL}(\textbf{s})/\textbf{P}\) where \(\textrm{GL}(\textbf{s})\) is a strict diagonal ind-group defined by a supernatural number \(\textbf{s}\) and \(\textbf{P}\) is a parabolic ind-subgroup of \(\textrm{GL}(\textbf{s})\). We construct an explicit exhaustion of \(\textrm{GL}(\textbf{s})/\textbf{P}\) by finite-dimensional partial flag varieties. As an application, we characterize all locally projective \(\textrm{GL}(\infty )\)-homogeneous spaces, and some direct products of such spaces, which are \(\textrm{GL}(\textbf{s})\)-homogeneous for a fixed \(\textbf{s}\). The very possibility for a \(\textrm{GL}(\infty )\)-homogeneous space to be \(\textrm{GL}(\textbf{s})\)-homogeneous for a strict diagonal ind-group \(\textrm{GL}(\textbf{s})\) arises from the fact that the automorphism group of a \(\textrm{GL}(\infty )\)-homogeneous space is much larger than \(\textrm{GL}(\infty )\).