# Subgroups of $Spin(7)$ or $SO(7)$ with Each Element Conjugate to Some Element of $G_{2}$ and Applications to Automorphic Forms

### Gaëtan Chenevier

CNRS, Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France

## Abstract

As is well-known, the compact groups $Spin(7)$ and $SO(7)$ both have a single conjugacy class of compact subgroups of exceptional type $G_{2}$. We first show that if $Γ$ is a subgroup of $Spin(7)$, and if each element of $Γ$ is conjugate to some element of $G_{2}$, then $Γ$ itself is conjugate to a subgroup of $G_{2}$. The analogous statement for $SO(7)$ turns out be false, and our main result is a classification of all the exceptions. They are the following groups, embedded in each case in $SO(7)$ in a very specific way: $GL_{2}(Z/3Z)$, $SL_{2}(Z/3Z)$, $Z/4Z×Z/2Z$, as well as the nonabelian subgroups of $GO_{2}(C)$ with compact closure, similitude factors group ${±1}$, and which are not isomorphic to the dihedral group of order 8. More generally, we consider the analogous problems in which the Euclidean space is replaced by a quadratic space of dimension 7 over an arbitrary field. This type of questions naturally arises in some formulation of a converse statement of Langlands' global functoriality conjecture, to which the results above have thus some applications. Moreover, we give necessary and sufficient local conditions on a cuspidal algebraic regular automorphic representation of $GL_{7}$ over a totally real number field so that its associated $ℓ$-adic Galois representations can be conjugate into $G_{2}(Q_{ℓ} )$. We provide 11 examples over $Q$ which are unramified at all primes.

## Cite this article

Gaëtan Chenevier, Subgroups of $Spin(7)$ or $SO(7)$ with Each Element Conjugate to Some Element of $G_{2}$ and Applications to Automorphic Forms. Doc. Math. 24 (2019), pp. 95–161

DOI 10.4171/DM/676