Visible Actions on Complex Manifolds,
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 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'/HG' 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.

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© Toshiyuki Kobayashi