Möb Homogeneous Analytic Hilbert Modules over the Bidisc

An analytic Hilbert module $\mathcal{H}$ over the polynomial ring, consisting of holomorphic functions over the bidisc, is said to be M\"ob-homogeneous if the corresponding pair of multiplication operators is homogeneous with respect to the diagonal action of the group $\{(\varphi,\varphi): \varphi \in \mbox{M\"ob}\} \cong \mbox{M\"ob}$. In this article, we construct three families of mutually unitarily inequivalent M\"ob-homogeneous analytic Hilbert modules, distinct from the family of weighted Bergman modules over the bidisc. We further show that none of the reproducing kernels in one of these families induces a K\"ahler--Einstein metric on the bidisc.