Nearly holomorphic automorphic forms on $\mathrm{SL}_2$

TL;DR

This paper explores nearly holomorphic automorphic forms on reductive groups over $\,\mathbb{Q}$, extending classification results to the adelic setting for Hilbert modular forms and connecting to prior work on elliptic and Siegel modular forms.

Contribution

It generalizes the classification of nearly holomorphic automorphic forms to the adelic framework for Hilbert modular forms, building on previous elliptic and Siegel modular form results.

Findings

Classification of global representations in nearly holomorphic Hilbert modular forms

Extension of elliptic modular form results to adelic setting

Connection to previous classifications by Pitale, Saha, and Schmidt

Abstract

We define the space of nearly holomorphic automorphic forms on a connected reductive group $G$ over $\mathbb{Q}$ such that the homogeneous space $G(\mathbb{R})^1/ K_\infty^\circ$ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's study, there are the classification of indecomposable $(\mathfrak{g},K_\infty)$-modules which occur in the space of nearly holomorphic elliptic modular forms and Siegel modular forms of degree $2$. This paper studies global representations of the adele group $G(\mathbb{A}_\mathbb{Q})$ which occur in the space of nearly holomorphic Hilbert modular forms. In the case of elliptic modular forms, the result of this paper is an adelization of Pitale, Saha and Schmidt's result.