Weyl invariant $E_8$ Jacobi forms

We investigate $W(E_8)$-invariant Jacobi forms which are the Jacobi forms invariant under the action of the Weyl group of the root system $E_8$. This type of Jacobi forms has applications in mathematics and physics, but very little has been known about its structure. In this paper we show that the bigraded ring of weak $W(E_8)$-invariant Jacobi forms is not a polynomial algebra over $C$ and prove that every $W(E_8)$-invariant Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent holomorphic Jacobi forms introduced by Sakai with coefficients which are meromorphic $SL_2(Z)$ modular forms. The latter result implies that the graded ring of weak $W(E_8)$-invariant Jacobi forms of fixed index is a free module over the ring of $SL_2(Z)$ modular forms and the number of generators can be calculated by a generating series. We also determine and construct all generators of small index. These results extend Wirthm\"{u}ller's theorem proved in 1992 to the last open case.