Paul, is there a representation theoretic characterization of the space of harmonic weak mass forms on the upper half plane? (They are functions on Shimura curves, though not necessarily L^2.)

Was thinking out the implication of the term "being the holomorphic part". The generating function of Hurwitz class numbers, as a mock modular form, is associated to Zagier's Eisenstein series of weight 3/2. recently noticed that some geometric generating functions are the ``holomorphic part" of some interesting Eisenstein series on Siegel upper half plane. Possibly some Eisenstein series contribute to a subset of the imagined generalization of harmonic weak mass form.

Thanks a lot. (For problems related to nonproper varieties and noncompact manifolds, I often wonder whether the theorems and constructions for compact varieties/manifolds would still work.)