Multiple flag ind-varieties with finitely 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 $X_1\times \cdots \times X_l$ 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 $l\ge 4$, there always are infinitely many G-orbits on $X_1\times \cdots \times X_l$.