A generalized Cartan decomposition for the double coset space U(n_1) x U(n_2) x U(n_3)) U(n) / U(p) x U(q)

Abstract

Motivated by recent developments on visible actions on complex manifolds, we raise a question whether or not the multiplication of three subgroups $L$, $G^{\prime }$ and $H$ surjects a Lie group $G$ in the setting that $G/H$ carries a complex structure and contains $G^{\prime }/G^{\prime }\cap H$ 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 $L{G}^{\prime }H=G$, or equivalently, the real generalized flag variety $G^{\prime }/H\cap G^{\prime }$ meets every $L$-orbit on the complex generalized flag variety $G/H$ in the setting that $(G,G^{\prime })=(U(n),O(n))$. For such pairs $(L,H)$, we introduce a \textit{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.

References in corpus (2)

• Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs
• Visible actions on symmetric spaces

Cited by in corpus (5)

• Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs
• Propagation of multiplicity-freeness property for holomorphic vector bundles
• Gamma Factors of Distinguished Representations of GL_n(C)
• On the Cartan Decomposition for Classical Random Matrix Ensembles
• Multiplicity-free representations and coisotropic actions of certain nilpotent Lie groups over quasi-symmetric Siegel domains