Description of infinite orbits on multiple flag varieties of type A (Representation Theory and Related Areas)

Abstract

Let G be a reductive group, P be its parabolic subgroup, and H be a closed subgroup of G. There are several studies on the orbit decomposition of the flag variety G/P by the H-action, and these studies are expected to play an important role in various problems such as branching problem of G with respect to H. In thib note, we focus on explicit descriptions of the orbit decomposition of a multiple flag variety (Gtimes Gtimes cdots times G)/(P_{1} times P_{2} times cdots times P_{m}) by the diagonal action of G. Now, let G be a general linear group on an algebraically closed field with characteristic 0. Magyar-Weyman-Zelevinsky proved that there are only finitely many orbits only if mleq 3. Furthermore, they also classificd all tuples (P_{1}, P2, ..., P_{m}) of parabolic subgroups where the number of orbits are finite, and gave explicit orbit decompositions for these cases. The aim of this note is to give an explicit description of the orbit decomposition for mgeq 4, the case where infinitely many orbits exist.