Rankin--Cohen brackets on Hermitian Jacobi forms and the adjoint of some linear maps

Abstract

Given a fixed Hermitian Jacobi cusp form, we define a family of linear operators between spaces of Hermitian Jacobi cusp forms using Rankin--Cohen brackets. We compute the adjoint maps of such a family with respect to the Petersson scalar product. The Fourier coefficients of the Hermitian Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Hermitian Jacobi cusp forms.