Weyl invariant E_8 Jacobi forms 2

In the first talk I have proved that the ring of Weyl invariant E_8 weak Jacobi forms of fixed index is a free module over the ring of SL(2,Z) modular forms and the number of generators is known. In this talk, I will determine and construct generators of small index. To do this, I will introduce two approaches. The first one relies heavily on the differential operators. From a Jacobi form of negative weight with indicated q^0-term, one builds a system of linear equations whose solution implies the existence of the given Jacobi form. The second is based on the pull-backs from E_8 Jacobi forms to classical Jacobi forms. The two approaches would be also useful to study the ring of Jacobi forms for other lattices.