$\textrm{Spin}(7)$ Is Unacceptable

Abstract

We classify to some extent the pairs of group morphisms $\Gamma \rightarrow \textrm{Spin}(7)$ which are element-conjugate but not globally conjugate. As an application, we study the case where $\Gamma $ is the Weil group of a p-adic local field, which is relevant to the recent approach to the local Langlands correspondence for $\textrm{G}_2$ and $\textrm{PGSp}_6$ in Gan and Savin (Forum Math Pi 11:e28, 2023). As a second application, we improve some result in Kret and Shin (J Eur Math Soc 25(1):75–152, 2023) about $\textrm{GSpin}_7$-valued Galois representations.

Funding

CNRS(ANR-19-CE40-0015-02 (COLOSS)); Singapore Government MOE(R-146-000-320-114)