Real group orbits on flag ind-varieties of SL (∞, C)

Abstract

We consider the complex ind-group $G=SL (\infty, \mathbb{C})$ and its real forms $G^0=SU(\infty,\infty)$, $SU(p,\infty)$, $SL(\infty,\mathbb{R})$, $SL(\infty,\mathbb{H})$. Our main object of study are the $G^0$-orbits on an ind-variety $G/P$ for an arbitrary splitting parabolic ind-subgroup $P \subset G$, under the assumption that the subgroups $G^0 \subset G$ and $P \subset G$ are aligned in a natural way. 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 close $G^0$-orbits on $G/P$ in the case of infinitely many $G^0$-orbits.