Visible actions and criteria for multiplicity-freeness of representations of Heisenberg groups

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 $σ$ such that $σ|_S = \operatorname{id}_S$. 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.