Rankin-Cohen operators for Jacobi and Siegel forms

Abstract

Far any non-negative integer nu we construct esplicitly [nu/2] + 1 independent covariant bilinear differential operators from J(k,m)xJ(k',m') to J(k+k'+nu,m+m'). As an application we construct a covariant bilinear differential operator mapping S-k((2))xS(k')((2)) to S-k+k'+c((2)). Here J(k,m) denotes the space of Jacobi forms of weight k and index m and S-k((2)) 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. (C) 1998 Academic Press.