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- W2983617291 abstract "A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $sigma $ such that $sigma |_S = operatorname{id}_S$. Let $G$ be the Heisenberg group and $H$ a non-trivial connected closed subgroup of $G$. We prove that any complex homogeneous space $D = G^{mathbb{C}}/H^{mathbb{C}}$ admits a strongly visible $L$-action, where $L$ stands for a connected closed subgroup of $G$ explicitly constructed through a co-exponential basis of $H$ in $G$. This leads in turn that $G$ itself acts strongly visibly on $D$. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic diffeomorphism and a totally real submanifold $S$, for which the dimension depends upon the dimensions of $G$ and $H$. As a direct application, our geometric results provide a proof of various multiplicity-free theorems on continuous representations on the space of holomorphic sections on $D$. Moreover, we also generate as a consequence, a geometric criterion for a quasi-regular representation of $G$ to be multiplicity-free." @default.
- W2983617291 created "2019-11-22" @default.
- W2983617291 creator A5011037105 @default.
- W2983617291 creator A5020359848 @default.
- W2983617291 date "2019-11-09" @default.
- W2983617291 modified "2023-09-27" @default.
- W2983617291 title "Visible actions and criteria for multiplicity-freeness of representations of Heisenberg groups." @default.
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- W2983617291 hasPublicationYear "2019" @default.
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