Quasimodular forms and automorphic pseudodifferential operators of mixed weight

Abstract

Jacobi-like forms for a discrete subgroup $\Gamma$ of $SL(2, \mathbb R)$ are formal power series which generalize Jacobi forms, and they are in one-to-one correspondence with automorphic pseudodifferential operators for $\Gamma$. The well-known Cohen–Kuznetsov lifting of a modular form f provides a Jacobi-like form and therefore an automorphic pseudodifferential operator associated to f. Given a pair $(\lambda , \mu )$ of integers, automorphic pseudodifferential operators can be extended to those of mixed weight. We show that each coefficient of an automorphic pseudodifferential operator of mixed weight is a quasimodular form and prove the existence of a lifting of Cohen–Kuznetsov type for each quasimodular form.