Weyl invariant E8 Jacobi forms

We investigate the Jacobi forms for the root system E-8 invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov-Witten theory and string theory. In 1992, Wirthmuller proved that the space of Jacobi forms for any irreducible root system not of type E-8 is a polynomial algebra. But very little has been known about the case of E-8. In this paper we show that the bigraded ring of Weyl invariant E-8 Jacobi forms is not a polynomial algebra and prove that every such Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent Jacobi forms introduced by Sakai with coefficients which are meromorphic SL2(Z) modular forms. The latter result implies that the space of Weyl invariant E-8 Jacobi forms of fixed index is a free module over the ring of SL2(Z) modular forms and that the number of generators can be calculated by a generating series. We determine and construct all generators of small index. These results give a proper extension of the Chevalley type theorem to the case of E-8.