Weyl invariant $E_8$ Jacobi forms 1

For the lattice constructed from the classical root system R, Wirthmuller defined Jacobi forms invariant under the Weyl group W(R). In 1992, Wirthmuller proved that the bigraded ring of W(R)-invariant weak Jacobi forms is a polynomial algebra over the ring of SL(2, Z) modular forms except the root system E_8. It is still an open problem how to extend the Wirthmuller's theorem to the case R=E_8. Weyl invariant E_8 Jacobi forms has applications in mathematics and physics, but very little has been known about its structure. In two talks, I will present an explicit description of the ring of W(E_8)-invariant Jacobi forms.
In this talk, I will first give a brief overview of Weyl invariant Jacobi forms and Wirthmuller's structure theorem. Then I will introduce a proper extension of Wirthmuller's theorem to the case of E_8 and show that the ring of W(E_8)-invariant weak Jacobi forms is not a polynomial algebra.