Toeplitz operators on the proper images of bounded symmetric domains

Let Ω be a bounded symmetric domain in Cⁿ and f :Ω → Ω^ be a proper holomorphic mapping factored by (automorphisms) a finite complex reflection group $G.$ We define an appropriate notion of the Hardy space H²(Ω⁾ on Ω^ which can be realized as a closed subspace of an L²-space on the {S}ilov boundary of Ω^. We study various algebraic properties of Toeplitz operators (such as the finite zero product property, commutative and semi-commutative property etc.) on H²(Ω⁾. We prove a Brown-Halmos type characterization for Toeplitz operators on H²(Ω⁾, where Ω^ is an image of the open unit polydisc in Cⁿ under a proper holomorphic mapping factored by an irreducible finite complex reflection group.