Subgroups of $\mathrm{Spin}(7)$ or $\mathrm{SO}(7)$ with Each Element Conjugate to Some Element of ${\mathrm{G}}_2$ and Applications to Automorphic Forms

Abstract

As is well-known, the compact groups $\mathrm{Spin}(7)$ and $\mathrm{SO}(7)$ both have a single conjugacy class of compact subgroups of exceptional type ${\mathrm{G}}_2$. We first show that if $\Gamma$ is a subgroup of $\mathrm{Spin}(7)$, and if each element of $\Gamma$ is conjugate to some element of ${\mathrm{G}}_2$, then $\Gamma$ itself is conjugate to a subgroup of $G_2$. The analogous statement for $\mathrm{SO}(7)$ 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 $\mathrm{SO}(7)$ in a very specific way: ${\mathrm{GL}}_2(\mathbb{Z}/3\mathbb{Z})$, ${\mathrm{SL}}_2(\mathbb{Z}/3\mathbb{Z})$, $\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, as well as the nonabelian subgroups of ${\mathrm{GO}}_2(\mathbb{C})$ with compact closure, similitude factors group $\{\pm 1\}$, 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 ${\mathrm{GL}}_7$ over a totally real number field so that its associated $\ell$-adic Galois representations can be conjugate into ${\mathrm{G}}_2(\overline{{\mathbb{Q}}_{\ell }})$. We provide 11 examples over $\mathbb{Q}$ which are unramified at all primes.