Multiple flag ind-varieties with finitely many orbits
Lucas Fresse, Ivan Penkov

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 be one of the ind-groups , , , and be an arbitrary set of splitting parabolic subgroups of . We determine all such sets with the property that acts with finitely many orbits on the ind-variety where . In the case of a finite-dimensional classical linear algebraic group , 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 , the condition that acts on with finitely many orbits is a rather restrictive condition on the pair . We describe this condition explicitly. Using this result, we tackle the most interesting case where , and present the answer in the form of a table. For , there always are…
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