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

Abstract

Let us consider, for $n\ge 3$, the Cartan domain ${\mathrm{D}}_n^{\mathrm{IV}}$ of type IV. On the weighted Bergman spaces ${\mathcal{A}}_{\lambda }^2({\mathrm{D}}_n^{\mathrm{IV}})$ we study the problem of the existence of commutative $C^{\ast }$-algebras generated by Toeplitz operators with special symbols. We focus on the subgroup $\mathrm{SO}(n)\times \mathrm{SO}(2)$ of biholomorphisms of ${\mathrm{D}}_n^{\mathrm{IV}}$ that fix the origin. The $\mathrm{SO}(n)\times \mathrm{SO}(2)$-invariant symbols yield Toeplitz operators that generate commutative $C^{\ast }$-algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under $\mathrm{SO}(2)$. We compute the moment map ${\mu }^{\mathrm{SO}(2)}$ for the $\mathrm{SO}(2)$-action on ${\mathrm{D}}_n^{\mathrm{IV}}$ considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form $a=f\circ {\mu }^{\mathrm{SO}(2)}$, denoted by $L^{\infty }({\mathrm{D}}_n^{\mathrm{IV}}{)}^{{\mu }^{\mathrm{SO}(2)}}$, yield Toeplitz operators that generate commutative $C^{\ast }$-algebras. Spectral integral formulas for these Toeplitz operators are also obtained.