On Homogeneous Spaces for Diagonal Ind-Groups

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