The Heisenberg group action on the Siegel domain and the structure of Bergman spaces

TL;DR

This paper explores the action of the Heisenberg group on the Siegel domain, decomposes associated Bergman spaces into Fock spaces, and shows that certain Toeplitz operator algebras are commutative and isomorphic to very slowly oscillating functions.

Contribution

It introduces a natural coordinate system for the Siegel domain adapted to the Heisenberg group, enabling new decompositions of Bergman spaces and analysis of Toeplitz operator algebras.

Findings

Decomposition of the domain into coordinates adapted to $ ext{H}_n$

Decomposition of Bergman spaces as direct integrals of Fock spaces

The Toeplitz algebra is commutative and isomorphic to $ ext{VSO}( ext{R}_+)$

Abstract

We study the biholomorphic action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \geq 1$). Such $\mathbb{H}_n$-action allows us to obtain decompositions of both $D_{n+1}$ and the weighted Bergman spaces $\mathcal{A}^2_\lambda(D_{n+1})$ ($\lambda > -1$). Through the use of symplectic geometry we construct a natural set of coordinates for $D_{n+1}$ adapted to $\mathbb{H}_n$. This yields a useful decomposition of the domain $D_{n+1}$. The latter is then used to compute a decomposition of the Bergman spaces $\mathcal{A}^2_\lambda(D_{n+1})$ ($\lambda > -1$) as direct integrals of Fock spaces. This effectively shows the existence of an interplay between Bergman spaces and Fock spaces through the Heisenberg group $\mathbb{H}_n$. As an application, we consider $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ the $C^*$-algebra acting on the weighted Bergman space $\mathcal{A}^2_\lambda(D_{n+1})$ ($\lambda > -1$) generated by Toeplitz operators whose symbols belong to $L^\infty(D_{n+1})^{\mathbb{H}_n}$ (essentially bounded and $\mathbb{H}_n$-invariant). We prove that $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ is commutative and isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$ (very slowly oscillating functions on $\mathbb{R}_+$), for every $\lambda > -1$ and $n \geq 1$.