Multiple flag ind-varieties with finitely many orbits

Abstract

Let G be one of the ind-groups GL(∞), Ø(∞), Sp(∞), and P 1 , . . . , 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 × • • • × 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 × 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 ≥ 4, there always are infinitely many G-orbits on X 1 × • • • × X ℓ .