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 $$\mathcal {A}^2_\lambda (\mathrm {D}_n^{\mathrm {IV}})$$ we study the problem of the existence of commutative $$C^*$$ -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 $$C^*$$ -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 $$C^*$$ -algebras. Spectral integral formulas for these Toeplitz operators are also obtained.