Motivated by the notion of ''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\G/H, for which there has been no general theory in the non-symmetric case.
Our geometric results provide a unified proof of various multiplicity-free theorems in representation theory of general linear groups.
© Toshiyuki Kobayashi