Jacobi Forms of Affine Weight in Higher Cogenus and Nearly Holomorphic Functions
Jan Feldmann, Martin Raum
TL;DR
The paper extends the Ibukiyama–Kyomura framework to Jacobi forms with affine weight for arbitrary cogenus $h$ by using covariant differential operators arising from differential geometry, avoiding heavy explicit calculations. It establishes a direct-sum decomposition of vector-valued Jacobi forms $\mathrm{J}_{(k,s),m}(\Γ)$ into classical scalar Jacobi spaces $\mathrm{J}_{k+\ell,m}(\Γ)$ with multiplicities $\binom{s-\ell+h-1}{h-1}$ under the condition $k>h/2$ and invertible half-integral index $m$. A holomorphic projection for nearly holomorphic Jacobi forms is developed, yielding a structure theorem that mirrors Shimura’s results and provides a robust, covariance-respecting approach. The decomposition is achieved through a combination of Frobenius reciprocity for homogeneous vector bundles and a covariant holomorphic projection, providing a unified framework for understanding Fourier–Jacobi expansions in higher cogenus and informing the Fourier-Jacobi theory of Siegel modular forms. The work culminates in a general, differential-geometric method to relate vector-valued Jacobi forms to classical ones, with potential applications to Siegel modular form theory and related lifting correspondences.
Abstract
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.