$\mathrm{Spin}(7)$ is unacceptable

Abstract

We classify the pairs of group morphisms $\Gamma \to \mathrm{Spin}(7)$ 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 ${\mathrm{G}}_2$ and ${\mathrm{PGSp}}_6$ by Gan and Savin. As a second application, we improve some result of Kret and Shin about ${\mathrm{GSpin}}_7$-valued Galois representations.

42 pages

References in corpus (2)

• Subgroups of Spin(7) or SO(7) with each element conjugate to some element of G_2, and applications to automorphic forms
• The Local Langlands Conjecture for $G_2$