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

TL;DR

This paper demonstrates that complex homogeneous spaces related to Heisenberg groups admit strongly visible actions, leading to new multiplicity-free theorems for certain continuous representations.

Contribution

It introduces explicit constructions of strongly visible actions on complex homogeneous spaces of Heisenberg groups, providing geometric criteria for multiplicity-freeness of associated representations.

Findings

Heisenberg group actions are strongly visible on complex homogeneous spaces.

Explicit anti-holomorphic diffeomorphisms and totally real submanifolds are constructed.

New multiplicity-free theorems for holomorphic sections and quasi-regular representations are proved.

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.