Commutative algebras of Toeplitz operators and Lagrangian foliations

Let $D$ be a homogeneous bounded domain of ${\mathbb{C}}^n$ and $\mathcal{A}$ a set of (anti–Wick) symbols that defines a commutative algebra of Toeplitz operators on every weighted Bergman space of $D$. We prove that if $\mathcal{A}$ is rich enough, then it has an underlying geometric structure given by a Lagrangian foliation.