Weyl invariant $E_8$ Jacobi forms

In 1992, Wirthmüller proved that for the lattice constructed from a classical root system (except E_8), the corresponding space of Jacobi forms which are invariant under the Weyl group is a polynomial algebra over the ring of modular forms. In the talk we will focus on discussing the remaining case E_8. We will show that the space of Weyl invariant E_8 Jacobi forms is not a polynomial algebra and present a proper extension of Wirthmüller's theorem in the case. We will also give the generators of indices 2, 3, 4.