Commutants of Toeplitz operators with separately radial polynomial symbols on the Fock space

My talk concerns commuting Toeplitz operators on the Fock space $\mathcal{F}^2_{n}$. Let $\varphi$ be a separately radial polynomial in $z$ and $\bar{z}$ in $\mathbb{C}^n$. Then the Toeplitz operator $T_{\varphi}$ is diagonal with respect to the standard orthonormal basis of $\mathcal{F}^2_{n}$. We obtain a characterization of polynomially bounded functions $\psi$ for which $T_{\psi}$ commutes with $T_{\varphi}$. Substantially different from the radial case, the characterization depends highly on the behavior of the polynomial $\varphi$. I will discuss several examples and consequences of our result. This is joint work with Amila Appuhamy