Weyl invariant Jacobi forms: A new approach

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