Brown-Halmos type Theorems on the proper images of bounded symmetric domains

Let $\mathrm{\Omega }\subseteq {\mathbb{C}}^n$ be a bounded symmetric domain and $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq {\mathbb{C}}^n$ be a proper holomorphic mapping which is factored by a finite complex reflection group $G\mathrm{.}$ We identify a family of reproducing kernel Hilbert spaces on ${\mathrm{\Omega }}^{\mathrm{\prime }}$ arising naturally from the isotypic decomposition of the regular representation of $G$ on the Hardy space $H^2(\mathrm{\Omega })\mathrm{.}$ Each element of this family can be realized as a closed subspace of some $L^2$-space on the Šilov boundary of ${\mathrm{\Omega }}^{\mathrm{\prime }}$. The reproducing kernel Hilbert space associated to the sign representation of $G$ is the Hardy space $H^2({\mathrm{\Omega }}^{\mathrm{\prime }})\mathrm{.}$ We establish a Brown-Halmos type characterization for the Toeplitz operators on $H^2({\mathrm{\Omega }}^{\mathrm{\prime }}),$ where ${\mathrm{\Omega }}^{\mathrm{\prime }}$ is the image of the open unit polydisc ${\mathbb{D}}^n$ in ${\mathbb{C}}^n$ under a proper holomorphic mapping factored by the finite complex reflection group $G(m,p,n)\mathrm{.}$ Moreover, we prove various multiplicative properties of Toeplitz operators on $H^2({\mathrm{\Omega }}^{\mathrm{\prime }})$, where ${\mathrm{\Omega }}^{\mathrm{\prime }}$ is a proper holomorphic image of a bounded symmetric domain.