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It appears that mock modular forms are currently defined as the ``holomorhic part" of a harmonic weak Mass form on the upper half plane (with regard to certain modular subgroup).

Is there any generalization of this concept to higher dimensional Shimura varieties?

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  • $\begingroup$ A good, obvious question. An elementary viewpoint will not be effective... :) Growth issues versus tame growth are not trivial, nor are representation-type issues. :) Was there a specific issue in mind> $\endgroup$
    – paul garrett
    Commented May 13, 2014 at 0:37
  • $\begingroup$ 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. $\endgroup$
    – John
    Commented May 13, 2014 at 8:16
  • $\begingroup$ 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.) $\endgroup$
    – John
    Commented May 13, 2014 at 9:45

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