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

Abstract

We study the biholomorphic action of the Heisenberg group ${\mathbb{H}}_n$ on the Siegel domain $D_{n+1}$ ($n\ge 1$). Such ${\mathbb{H}}_n$-action allows us to obtain decompositions of both $D_{n+1}$ and the weighted Bergman spaces ${\mathcal{A}}_{\lambda }^2(D_{n+1})$ ($\lambda >-1$). 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 ${\mathcal{A}}_{\lambda }^2(D_{n+1})$ ($\lambda >-1$) 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 $C^{\ast }$-algebra acting on the weighted Bergman space ${\mathcal{A}}_{\lambda }^2(D_{n+1})$ ($\lambda >-1$) 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 $n\ge 1$.