Rankin–Cohen Operators for Jacobi and Siegel Forms

Abstract

For any non-negative integer v we construct explicitly ⌊v/2⌋+1 independent covariant bilinear differential operators fromJk,m×Jk′,m′toJk+k′+v,m+m′. As an application we construct a covariant bilinear differential operator mappingS(2)k×S(2)k′toS(2)k+k′+v. HereJk,mdenotes the space of Jacobi forms of weightkand indexmandS(2)kthe space of Siegel modular forms of degree 2 and weightk. The covariant bilinear differential operators constructed are analogous to operators already studied in the elliptic case by R. Rankin and H. Cohen and we call them Rankin–Cohen operators.