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

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