Toeplitz operators and Hilbert modules on the symmetrized polydisc

When is the collection of $\mathsf{S}$-Toeplitz operators with respect to a tuple of commuting bounded operators $\mathsf{S}=(S_1,S_2,\dots ,S_{d-1},P)$, which has the symmetrized polydisc as a spectral set, non-trivial? The answer is in terms of powers of $P$ as well as in terms of a unitary extension. En route, Brown-Halmos relations are investigated. A commutant lifting theorem is established. Finally, we establish a general result connecting the $C^{\ast }$-algebra generated by the commutant of $\mathsf{S}$ and the commutant of its unitary extension $\mathsf{R}$.