The ring of weak Jacobi forms for $D_n$ root lattice

For every quadratic lattice one can defines the notion of weak Jacobi forms associated with this lattice. In 1992 K. Withmüller proved that spaces of Jacobi forms associated with root lattices (except E_8) have the structure of a free algebra over the ring of modular forms. However, his proof is very complicated. In my talk I plan to introduce some constructions that help to solve this problem in case D_8 and obtain generators of the corresponding algebra in an explicit way. The talk will be based on joint results with Valery Gritsenko. These results are preparing to publication.