The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators

Abstract

The Lie theoretic nature of the Rankin–Cohen brackets is here uncovered. These bilinear operations, which, among other purposes, were devised to produce a holomorphic automorphic form from any pair of such forms, are instances of SL ( 2 , R ) -equivariant holomorphic bi-differential operators on the upper half-plane. All of the latter are here characterized and explicitly obtained, by establishing their one-to-one correspondence with singular vectors in the tensor product of two sl ( 2 , C ) Verma modules. The Rankin–Cohen brackets arise in the generic situation where the linear span of the singular vectors of a given weight is one-dimensional. The picture is completed by the special brackets which appear for the finite number of pairs of initial lowest weights for which the above space is two-dimensional. Explicit formulæ for basis vectors in both situations are obtained and universal Lie algebraic objects subsuming all of them are exhibited. A few applications of these results and Lie theoretic approach are then considered. First, a generalization of the latter yields Rankin–Cohen type brackets for Hilbert modular forms. Then, some Rankin–Cohen brackets are shown to intertwine the tensor product of two holomorphic discrete series representations of SL ( 2 , R ) with another such representation occurring in the tensor product decomposition. Finally, the sought for precise relationship between the Rankin–Cohen brackets and Gordan's transvection processes of the nineteenth century invariant theory is unveiled.