Quasimodular forms, Jacobi-like forms, and pseudodifferential operators

We study various properties of quasimodular forms by using their connections with Jacobi-like forms and pseudodifferential operators. Such connections are made by identifying quasimodular forms for a discrete subgroup $\G$ of $SL(2, \bR)$ with certain polynomials over the ring of holomorphic functions of the Poincar\'e upper half plane that are $\G$-invariant. We consider a surjective map from Jacobi-like forms to quasimodular forms and prove that it has a right inverse, which may be regarded as a lifting from quasimodular forms to Jacobi-like forms. We use such liftings to study Lie brackets and Rankin-Cohen brackets for quasimodular forms. We also discuss Hecke operators and construct Shimura isomorphisms and Shintani liftings for quasimodular forms.