On Homogeneous Spaces for Diagonal Ind-Groups

Abstract

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