Abstract
Motivated by recent developments on visible actions on complex manifolds, we raise a question whether or not the multiplication of three subgroups L, G′ and H surjects a Lie group G in the setting that G/H carries a complex structure and contains G′/G′∩H as a totally real submanifold.
Particularly important cases are when G/L and G/H are generalized flag varieties, and we classify pairs of Levi subgroups (L,H) such that LG′H=G, or equivalently, the real generalized flag variety G′/H∩G′ meets every L-orbit on the complex generalized flag variety G/H in the setting that (G,G′)=(U(n),O(n)). For such pairs (L,H), we introduce a herringbone stitch method to find a generalized Cartan decomposition for the double coset space L$¥backslash$G/H, for which there has been no general theory in the non-symmetric case. Our geometric results provides a unified proof of various multiplicity-free theorems in representation theory of general linear groups.
