Workshop on Representation Theory and Prehomogeneous Vector Spaces (organized by Yumiko Hironaka, Iris Muller, Hiroyuki Ochiai, Hubert Rubenthaler, Fumihiro Sato), Institut de Recherche Mathématique Avancée (IRMA), Strasbourg, France, 11-14 September 2006.

Motivated by the notion of ''visible actions on complex manifolds'', we raise a question whether or not the multiplication of three subgroupsL,G' andHsurjects a Lie groupGin the setting thatG/Hcarries a complex structure and containsG'/G' ∩Has a totally real submanifold.Particularly important cases are when

G/LandG/Hare generalized flag varieties, and we classify pairs of Levi subgroups (L,H) such thatLG'H=G, or equivalently, the real generalized flag varietyG'/H∩G' meets everyL-orbit on the complex generalized flag varietyG/Hin the setting that (G,G') = (U(n),O(n)). For such pairs (L,H), we introduce aherringbone stitchmethod to find a generalized Cartan decomposition for the double coset spaceL\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