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Journal of Lie Theory 31 (2021), No. 3, 719--750
Copyright Heldermann Verlag 2021

Visible Actions and Criteria for Multiplicity-Freeness of Representations of Heisenberg Groups

Ali Baklouti
Dept. of Mathematics, Faculty of Science of Sfax, Sfax, Tunisia

Atsumu Sasaki
Dept. of Mathematics, Tokai University, Kanagawa, Japan
and: Dept. of Mathematics, Friedrich-Alexander-Universitšt, Erlangen, Germany


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.

Keywords: Visible action, slice, Heisenberg group, Heisenberg homogeneous space, multiplicity-free representation.

MSC: 22E25; 22E27.

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