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

We study the biholomorphic action of the Heisenberg group H n on the Siegel domain D n + 1 ( n ≥ 1 ). Such H n -action allows us to obtain decompositions of both D n + 1 and the weighted Bergman spaces A λ 2 (D n + 1) ( λ > - 1 ). Through the use of symplectic geometry we construct a natural set of coordinates for D n + 1 adapted to 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 A λ 2 (D n + 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 H n . As an application, we consider T (λ) (L ∞ (D n + 1) H n) the C ∗ -algebra acting on the weighted Bergman space A λ 2 (D n + 1) ( λ > - 1 ) generated by Toeplitz operators whose symbols belong to L ∞ (D n + 1) H n (essentially bounded and H n -invariant). We prove that T (λ) (L ∞ (D n + 1) H n) is commutative and isomorphic to VSO (R +) (very slowly oscillating functions on R + ), for every λ > - 1 and n ≥ 1 . [ABSTRACT FROM AUTHOR]