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

期刊论文
发表日期：

2024年8月15日

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Julio A. Barrera-Reyes,
Raúl Quiroga-Barranco

摘要

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