Rankin-Cohen Operators for Jacobi and Siegel Forms

For any non-negative integer v we construct explicitly ⌊v2⌋+1 independent covariant bilinear differential operators from Jk,m × Jk′,m′ to Jk+k′+v,m+m′ . As an application we construct a covariant bilinear differential operator mapping S (2) k ×S (2) k′ to S (2) k+k′+v. Here Jk,m denotes the space of Jacobi forms of weight k and index m and S (2) k the space of Siegel modular forms of degree 2 and weight k. 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. DAMTP-96-106 alg-geom/96mmnnn Partially supported by KOSEF 941-01100-001-2 and Basic Science BSRI 96-1431. Supported by the EPSRC and partially by PPARC and EPSRC (grant GR/J73322).