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Abstract

For any irreducible root system not of type $E_8$, Wirthm\"{u}ller proved in
1992 that the bigraded algebra of weak Jacobi forms invariant under the Weyl
group is a polynomial algebra. In this paper we give a new automorphic proof of
this result based on the general theory of Jacobi forms. We proved in a
previous paper that the space of weak Jacobi forms for $E_8$ is not a
polynomial algebra and every $E_8$ 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. In this paper
we further show that these coefficients are the quotients of some $SL_2(Z)$
modular forms by a certain power of a fixed $SL_2(Z)$ modular form which is
completely determined by the nine generators.