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.