Rankin–Cohen Brackets for Jacobi Forms and Tensor Products of Discrete Series Representations of the Jacobi Group

Explicit formulae for Rankin–Cohen brackets for Jacobi forms are obtained by pure Lie theoretic means. The Lie algebraic nature of these holomorphic bi-differential operators, which map pairs of Jacobi forms onto another such form, is unveiled, along with their relationship with the decompositions of tensor products of two (anti-)holomorphic discrete series representations of the Jacobi group $\mathfrak G$ into direct sums of similar representations. These decompositions, which turn out to be nonmultiplicity-free, are a problem of an independent interest, which is explicitly solved. Some of the obtained brackets are shown to be intertwiners, projecting the appropriate tensor product onto a direct summand. The first mentioned goal is reached by characterizing all the $\mathfrak G$-equivariant holomorphic bi-differential operators, through establishing a one-to-one correspondence between them and the null vectors in the tensor products of two Verma $\mathfrak g$-modules, for $\mathfrak g$ the complexified Lie algebra of $\mathfrak G$. A large class of pairs of such modules is considered, the remaining situations being relegated to a future work. A thorough analysis of the structure of the tensor products, based on Borho’s isomorphism, which provides a factorization of the universal enveloping algebra of $\mathfrak g$, yields explicit formulae for the sought after null vectors; they turn out to be completely determined by the null vectors in tensor products of three Verma $\mathfrak {sl}_2$-modules. The bi-differential operators associated with the obtained null vectors are then shown to be Rankin–Cohen brackets for Jacobi forms. The Lie theoretic approach developed here supplies a unifying framework within which results of previous investigations undertaken by different authors, using techniques pertaining to the realm of number theory, are recovered, provided with new interpretations and generalized. In particular, Böcherer’s generic dimensions are shown to be given by the above-mentioned multiplicities.