Function theoretic aspects of the symmetrized polydisc and generalization

We introduce Schur-Agler type class for the symmetrized polydisc ${\mathbb{G}}_d$ and establish a realization theorem for functions in this class. We state and prove an interpolation theorem on ${\mathbb{G}}_d$ with interpolating functions belonging to the associated Schur-Agler type class. Moreover, Toeplitz corona and extension theorems are established for ${\mathbb{G}}_d$. We also extend the realization, interpolation, Toeplitz corona and extension theorems to a more general symmetrized family of domains ${\mathrm{\Theta }}_d$.