Weyl invariant Jacobi forms: A new approach

Abstract

The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type E8. This approach is also applicable to E8. Even if the algebra of E8 Jacobi forms is known to be non-free, we still derive a new structure result.