Theta Functions Associated with the Affine Root Systems and the Elliptic Ruijsenaars Operators

We study a family of mutually commutative difference operators associated with the affine root systems. These operators act on the space of meromorphic functions on the Cartan subalgebra of the affine Lie algebra. We show that the space spanned by the characters of a fixed positive level is invariant under the action of these operators.