Representations of SL_2(R) and nearly holomorphic modular forms

TL;DR

This paper offers a new representation-theoretic proof of Shimura's structure theorem for nearly holomorphic modular forms on SL_2(R), using category O and automorphic form decomposition, with potential for generalization to other groups.

Contribution

It provides a novel proof of the structure theorem using representation theory, extending the approach to broader contexts beyond classical methods.

Findings

Decomposition of automorphic forms on SL_2(R) into indecomposable modules.

Application of category O to analyze nearly holomorphic modular forms.

Framework generalizes to other groups, as demonstrated in related work.

Abstract

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.