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Abstract

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, Wirthm\"{u}ller 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 SL(2,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 SL(2,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 $E_8$.