Toeplitz operators on quotient domains

Let $G$ be a finite pseudo-reflection group and $\Omega$ be a bounded domain in $\mathbb C^d$ which is $G$-invariant. The quotient domain $\Omega/G,$ is biholomorphically equivalent to a domain ${{\boldsymbol \theta}} (\Omega)$ where ${{\boldsymbol \theta}} : \Omega \to {{\boldsymbol \theta}}(\Omega)$ is a basic polynomial map. Prominent example of a quotient domain is the symmetrized polydisc $\mathbb G_d$ in $\mathbb C^d.$ In this case, the basic polynomial map is given by $z \to (s_1(z), s_2(z), \cdots s_d(z))$ from $\mathbb D^d$ (unit polydisc in $\mathbb C^d$) to $\mathbb G_d$ where $s_j(z)$ is the $j$-th elementary symmetric polynomial. We study properties of Toeplitz operators on weighted Bergman spaces on ${{\boldsymbol \theta}}(\Omega)$ by establishing a connection of them with Toeplitz operators on weighted Bergman spaces on $\Omega.$ Results on zero product problem and commuting pairs of Toeplitz operators will be explained. Representation theory of $G$ and projections to isotypic components play an important role in our results. (Joint work with Gargi Ghosh)