In this semi-expository note, we give a new proof of a structure theorem due
to Shimura for nearly holomorphic modular forms on the complex upper half
plane. Roughly speaking, the theorem says that the space of all nearly
holomorphic modular forms is the direct sum of the subspaces obtained by
applying appropriate weight-raising operators on the spaces of holomorphic
modular forms and on the one-dimensional space spanned by the weight 2 nearly
holomorphic Eisenstein series.
While Shimura's proof was classical, ours is representation-theoretic. We
deduce the structure theorem from a decomposition for the space of n-finite
automorphic forms on SL_2(R). To prove this decomposition, we use the mechanism
of category O and a careful analysis of the various possible indecomposable
submodules. It is possible to achieve the same end by more direct methods, but
we prefer this approach as it generalizes to other groups.
This note may be viewed as the toy case of our paper ["Lowest weight modules
of Sp_4(R) and nearly holomorphic Siegel modular forms"], where we prove an
analogous structure theorem for vector-valued nearly holomorphic Siegel modular
forms of degree two.Comment: 13 page
