Commuting Toeplitz operators and moment maps on Cartan domains of type III

Let $D^{III}_n$ and $\mathscr{S}_n$ be the Cartan domains of type III that consist of the symmetric $n \times n$ complex matrices $Z$ that satisfy $Z\overline{Z} < I_n$ and $\mathrm{Im}(Z) > 0$, respectively. For these domains, we study weighted Bergman spaces and Toeplitz operators acting on them. We consider the Abelian groups $\mathbb{T}$, $\mathbb{R}_+$ and $\mathrm{Symm}(n,\mathbb{R})$ (symmetric $n \times n$ real matrices), and their actions on the Cartan domains of type III. We call the corresponding actions Abelian Elliptic, Abelian Hyperbolic and Parabolic. The moment maps of these three actions are computed and functions of them (moment map symbols) are used to construct commutative $C^*$-algebras generated by Toeplitz operators. This leads to a natural generalization of known results for the unit disk. We also compute spectral integral formulas for the Toeplitz operators corresponding to the Abelian Elliptic and Parabolic cases.