Pseudodifferential Jacobi forms and Geometric Rankin-Cohen Brackets

Abstract

Cohen, Manin, and Zagier recovered the Rankin-Cohen bracket for modular forms from an action of the modular group on pseudodifferential operators whose coefficients are holomorphic functions on the Poincaré upper half plane. We investigate pseudodifferential operators on the Jacobi upper half space with respect to the elliptic variable instead of the modular variable typically considered. We introduce a family of actions of the Jacobi group and show that a space of invariant pseudodifferential operators is isomorphic to the space of Jacobi forms by producing an equivariant map. Our construction arises from the explicit action of a Casimir operator for the complexified Lie algebra of the real Jacobi Lie group. As an application, we identify new families of Rankin-Cohen brackets with geometric origin indexed by a complex parameter. In particular, we isolate a subvariety of lines of Rankin-Cohen brackets in each degree of expected dimension $1$ reflecting the geometry of the Jacobi upper half space.