Automorphic Pseudodifferential Operators

Abstract

The theme of this paper is the correspondence between classical modular forms and pseudodifferential operators (ΨDO's) which have some kind of automorphic behaviour. In the simplest case, this correspondence is as follows. Let Γ be a discrete subgroup of PSL 2(ℝ) acting on the complex upper half-plane H in the usual way, and f(z) a modular form of even weight k on Γ. Then there is a unique lifting from f to a Γ-invariant ΨDO with leading term f(z)∂-k/2, where ∂ is the differential operator $$ \frac{d}{{dz}} $$ . This lifting and the fact that the product of two invariant ΨDO's is again an invariant ΨDO imply a non-commutative multiplicative structure on the space of all modular forms whose components are scalar multiples of the so-called Rankin-Cohen brackets (canonical bilinear maps on the space of modular forms on Γ defined by certain bilinear combinations of derivatives; the definition will be recalled later). This was already discussed briefly in the earlier paper [Z], where it was given as one of several "raisons d'être" for the Rankin-Cohen brackets.