Finite-dimensional representations of the elliptic modular double

We investigate the kernel space of an integral operator $\mathrm M(g)$ depending on the "spin" $g$ and describing an elliptic Fourier transformation. The operator $\mathrm M(g)$ is an intertwiner for the elliptic modular double formed from a pair of Sklyanin algebras with the parameters $\eta$ and $\tau$, $\operatorname{Im}\tau>0$, $\operatorname{Im}\eta>0$. For two-dimensional lattices $g=n\eta+m\tau/2$ and $g=1/2+n\eta+m\tau/2$ with incommensurate $1,2\eta,\tau$ and integers $n,m>0$, the operator $\mathrm M(g)$ has a finite-dimensional kernel that consists of the products of theta functions with two different modular parameters and is invariant under the action of generators of the elliptic modular double.