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

Gaëtan Chenevier

Documenta mathematica, Tome 24 (2019), pp. 95-161

Résumé

As is well-known, the compact groups Spin(7) and SO(7) both have a single conjugacy class of compact subgroups of exceptional type G2​. We first show that if Γ is a subgroup of Spin(7), and if each element of Γ is conjugate to some element of G2​, then Γ itself is conjugate to a subgroup of G2​. 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: GL2​(Z/3Z), SL2​(Z/3Z), Z/4Z×Z/2Z, as well as the nonabelian subgroups of GO2​(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 GL7​ over a totally real number field so that its associated l-adic Galois representations can be conjugate into G2​(Ql​​). We provide 11 examples over Q which are unramified at all primes.

Détail

DOI : 10.4171/dm/676

Classification : 11F80, 11R39, 20G15, 20G41, 22C05
Mots-clés : Galois representations, automorphic forms, exceptional group G2​, subgroups of SO(7), Langlands conjectures