Real group orbits on flag ind-varieties of $\mathrm{SL}(\infty,\mathbb{C})$

TL;DR

This paper studies the action of real forms of the infinite-dimensional special linear group on ind-varieties of flags, characterizing orbit intersections, finiteness, and conditions for open and closed orbits.

Contribution

It provides a detailed analysis of real group orbits on flag ind-varieties of $ ext{SL}( ext{infinity}, ext{C})$, including orbit intersection properties and criteria for orbit finiteness and openness.

Findings

Intersection of orbits with finite-dimensional flag varieties is a single orbit.

Characterization of ind-varieties with finitely many orbits.

Criteria for existence of open and closed orbits.

Abstract

We consider the complex ind-group $G=\mathrm{SL}(\infty,\mathbb{C})$ and its real forms $G^0=\mathrm{SU}(\infty,\infty)$, $\mathrm{SU}(p,\infty)$, $\mathrm{SL}(\infty,\mathbb{R})$, $\mathrm{SL}(\infty,\mathbb{H})$. Our main objects of study are the $G^0$-orbits on an ind-variety $G/P$ for an arbitrary splitting parabolic ind-subgroup $P\subset G$. We prove that the intersection of any $G^0$-orbit on $G/P$ with a finite-dimensional flag variety $G_n/P_n$ from a given exhaustion of $G/P$ via $G_n/P_n$ for $n\to\infty$, is a single $(G^0\cap G_n)$-orbit. We also characterize all ind-varieties $G/P$ on which there are finitely many $G^0$-orbits, and provide criteria for the existence of open and closed $G^0$-orbits on $G/P$ in the case of infinitely many $G^0$-orbits.