Jacobi Forms of Affine Weight in Higher Cogenus and Nearly Holomorphic Functions

We describe Jacobi forms of vector-valued weights in terms of classical ones, extending previous results by Ibukiyama and Kyomura to the case of arbitrary cogenus. As in their result, our isomorphisms are given by holomorphic covariant differential operators. In contrast to previous work, however, we avoid explicit calculations, which we replace by general differential geometric arguments. In the process, we obtain a structure theorem on nearly holomorphic functions on the Jacobi upper half space.