  Let $G$ be a finite pseudoreflection group and $\Omega\subseteq \mathbb C^d$
be a bounded domain which is a $G$-space. We establish identities involving
Toeplitz operators on the weighted Bergman spaces of $\Omega$ and $\Omega/G$
using invariant theory and representation theory of $G.$ This, in turn,
provides techniques to study algebraic properties of Toeplitz operators on the
weighted Bergman space on $\Omega/G.$ We specialize on the generalized
zero-product problem and characterization of commuting pairs of Toeplitz
operators. As a consequence, more intricate results on Toeplitz operators on
the weighted Bergman spaces on some specific quotient domains (namely
symmetrized polydisc, monomial polyhedron, Rudin's domain) have been obtained.
