Counting Discrete, Level$1$, Quaternionic Automorphic Representations on $G_2$
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 study quaternionic automorphic representations on the exceptional group $G_2$. Using "hyperendoscopy" techniques from arXiv:1910.10800, we develop for quaternionic, $G_2$representations an analog of the EichlerSelberg trace formula for classical modular forms. We then use this together with some techniques of Chenevier, Renard, and Taïbi to compute dimensions of spaces of level$1$ quaternionic representations. On the way, we prove a JacquetLanglandsstyle result describing them in terms of classical modular forms and automorphic representations on the compactatinfinity form $G_2^c$. The main technical difficulty is that the quaternionic discrete series that quaternionic autmorphic representations are defined in terms of do not satisfy a condition of being "regular". Using some computations from arXiv:2010.02712, we show that this miraculously does not matter for specifically the case of quaternionic discrete series on $G_2$. We hope that this project serves as a guiding example for anyone interested in computing exact information about discreteatinfinity automorphic representations on arbitrary reductive groups instead of just classical ones.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.09313
 Bibcode:
 2021arXiv210609313D
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory;
 11F55 (primary) 11F70;
 11F72;
 11F75;
 20G41;
 22E50;
 22E55 (secondary)
 EPrint:
 23 pages, 1 figure, 1 table