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【ESI会议04/22】Rahul Dalal - Counting Level-1, Quaternionic Automorphic Reps on G_2
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https://www.youtube.com/watch?v=KbLEFpuymkw&list=PLjvY6sC_0lhIO0-DggR3l4Etr9m9H2dJ9&index=17 This talk was part of the Workshop on "Minimal Representations and Theta Correspondence" held at the ESI April 11 to 15, 2022. We count quaternionic automorphic representations on the exceptional group G2 by developing a G2 version of the classical Eichler-Selberg trace formula for holomorphic modular forms. There are three main technical points. First, quaternionic discrete series come in L-packets with non-quaternionic members and standard invariant trace formula techniques cannot easily distinguish between discrete series with real component in the same L-packet. Using the more modern stable trace formula resolves this issue. Second, quaternionic discrete series do not satisfy a technical condition of being "regular", so the trace formula can a priori pick up unwanted contributions from automorphic representations with non-tempered components at infinity---some work with real representation theory is required. Finally, we apply some tricks of Chenevier, Renard, and Taïbi for the level-1 case to avoid onerous computations on the geometric side.
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