A generalized Cartan decomposition for the double coset space $(U(n_1) \times U(n_2) \times U(n_3)) \backslash U(n) / (U(p) \times U(q))$

Abstract

Motivated by recent developments on visible action 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 $G' / G' \cap H$ as a totally real submanifold.

Paticularly 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 $G^{\prime} / \cap G^{\prime}$ 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.