The Heisenberg Group Action on the Siegel Domain and the Structure of Bergman Spaces

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 \(\mathcal {A}^2_\lambda (D_{n+1})\) (\(\lambda > -1\)). 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 \(\textrm{VSO}(\mathbb {R}_+)\) (very slowly oscillating functions on \(\mathbb {R}_+\)), for every \(\lambda > -1\) and \(n \ge 1\).