The ring of weak Jacobi forms for root lattices of D_n type

In 1992 K. Wirthmüller claimed in his article that the 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 contains some weak points, especially in the case of $D_n$ lattices. In my talk I will present an alternative proof of Wirthmüller's result for this type of lattices and, moreover, I will construct generators of corresponding algebras in an explicit way.