T. Kobayashi,

*A generalized cartan decomposition for the double coset
space* (*U*(*n*_{1})×*U*(*n*_{2})×*U*(*n*_{3}))\*U*(*n*)/(*U*(*p*)×*U*(*q*)), Journal of Mathematical Society of Japan **59** (2007), no. 3, 669-691,
math.RT/0607006..

Motivated by recent developments on 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 provides a unified proof of various multiplicity-free theorems in representation theory of general linear groups.

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The original publication is available at projecteuclid.org.

© Toshiyuki Kobayashi