
Journal of Lie Theory 31 (2021), No. 3, 719750 Copyright Heldermann Verlag 2021 Visible Actions and Criteria for MultiplicityFreeness of Representations of Heisenberg Groups Ali Baklouti Dept. of Mathematics, Faculty of Science of Sfax, Sfax, Tunisia ali.baklouti@fss.usf.tn Atsumu Sasaki Dept. of Mathematics, Tokai University, Kanagawa, Japan and: Dept. of Mathematics, FriedrichAlexanderUniversität, Erlangen, Germany atsumu@tokaiu.jp [Abstractpdf] 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 orbitpreserving antiholomorphic diffeomorphism $\sigma $ such that $\sigma _S = \operatorname{id}_S$. Let $G$ be the Heisenberg group and $H$ a nontrivial 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 coexponential 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 orbitpreserving antiholomorphic 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 multiplicityfree theorems on continuous representations on the space of holomorphic sections on $D$. Moreover, we also generate as a consequence, a geometric criterion for a quasiregular representation of $G$ to be multiplicityfree. Keywords: Visible action, slice, Heisenberg group, Heisenberg homogeneous space, multiplicityfree representation. MSC: 22E25; 22E27. [ Fulltextpdf (244 KB)] for subscribers only. 