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

TL;DR

This paper investigates when Toeplitz operators on certain symmetric domains generate commutative algebras, focusing on specific group actions and moment maps, and provides spectral formulas for these operators.

Contribution

It characterizes symbols leading to commutative Toeplitz operator algebras on Cartan domains of type IV using moment maps and symmetry considerations.

Findings

Invariant symbols under SO(n)×SO(2) produce commutative algebras.

Symbols composed with the moment map μ^{SO(2)} generate commutative Toeplitz algebras.

Spectral integral formulas for these Toeplitz operators are established.

Abstract

Let us consider, for $n \geq 3$, 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.