Multiple flag ind-varieties with finitely many orbits

TL;DR

This paper classifies sets of parabolic subgroups of infinite-dimensional classical groups where the group action on their product varieties has finitely many orbits, extending finite-dimensional results to the ind-group setting.

Contribution

It determines all such sets of parabolic subgroups for ind-groups, providing explicit conditions and a classification for the case of three factors, and showing infinite orbits for four or more.

Findings

Finite orbits occur only for specific pairs of parabolic subgroups in the ind-group setting.

Explicit classification of triples of parabolic subgroups with finitely many orbits.

For four or more factors, the action always has infinitely many orbits.

Abstract

Let $G$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$, and $P_1,\dots, P_l$ be an arbitrary set of $l$ splitting parabolic subgroups of $G$. We determine all such sets with the property that $G$ acts with finitely many orbits on the ind-variety $X_1\times\dots\times X_l$ where $X_i=G/P_i$. In the case of a finite-dimensional classical linear algebraic group $G$, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar-Weyman-Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for $l=2$, the condition that $G$ acts on $X_1\times X_2$ with finitely many orbits is a rather restrictive condition on the pair $P_1,P_2$. We describe this condition explicitly. Using this result, we tackle the most interesting case where $l=3$, and present the answer in the form of a table. For $l\geq 4$, there always are infinitely many G-orbits on $X_1\times \dots\times X_l$.