Counting Discrete, Level-$1$, Quaternionic Automorphic Representations on $G_2$

TL;DR

This paper develops a trace formula for quaternionic automorphic representations on the exceptional group G_2, enabling the computation of their dimensions and establishing links to classical modular forms.

Contribution

It introduces a general hyperendoscopy-based trace formula for automorphic representations, specializes it to G_2, and computes dimensions of level-1 quaternionic representations, extending classical results.

Findings

Derived an analog of the Eichler-Selberg trace formula for G_2

Computed dimensions of level-1 quaternionic automorphic representations

Established a Jacquet-Langlands-style correspondence for G_2 representations

Abstract

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm{GL}_2$. Here, we use "hyperendoscopy" techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$, eventually getting an analog of the Eichler-Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard, and Ta\"ibi to compute dimensions of spaces of level-$1$ quaternionic representations. On the way, we prove a Jacquet-Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$. The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being "regular". A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series. We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.