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 Möb-homogeneous if the corresponding pair of multiplication operators is homogeneous with respect to the diagonal action of the group $\{(\phi ,\phi ):\phi \in M\ddot{o}b\}\cong M\ddot{o}b$. In this article, we construct three families of mutually unitarily inequivalent Möb-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 Kähler–Einstein metric on the bidisc.