On homogeneous spaces for diagonal ind-groups

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