Toeplitz operators and group-moment coordinates for quasi-elliptic and quasi-hyperbolic symbols

For \(\mathbb {B}^n\) ${\mathbb{B}}^n$ the n-dimensional unit ball and \(D_n\) $D_n$ its Siegel unbounded realization, we consider Toeplitz operators acting on weighted Bergman spaces with symbols invariant under the actions of the maximal Abelian subgroups of biholomorphisms \(\mathbb {T}^n\) ${\mathbb{T}}^n$ (quasi-elliptic) and \(\mathbb {T}^n \times \mathbb {R}_+\) ${\mathbb{T}}^n\times {\mathbb{R}}_{+}$ (quasi-hyperbolic). Using geometric symplectic tools (Hamiltonian actions and moment maps) we obtain simple diagonalizing spectral integral formulas for such kinds of operators. Some consequences show how powerful the use of our differential geometric methods are.