Finite-dimensional representations of the elliptic modular double

Abstract

We investigate the kernel space of an integral operator M(g) depending on the “spin” g and describing an elliptic Fourier transformation. The operator M(g) is an intertwiner for the elliptic modular double formed from a pair of Sklyanin algebras with the parameters η and τ, Imτ > 0, Imη > 0. For two-dimensional lattices g = nη + mτ/2 and g = 1/2 + nη + mτ/2 with incommensurate 1, 2η, τand integers n,m > 0, the operator 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.