Subgroups of Spin(7)\mathrm{Spin}(7) or SO(7)\mathrm{SO}(7) with Each Element Conjugate to Some Element of G2\mathrm{G}_2 and Applications to Automorphic Forms

  • Gaëtan Chenevier

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

As is well-known, the compact groups Spin(7)\mathrm{Spin}(7) and SO(7)\mathrm{SO}(7) both have a single conjugacy class of compact subgroups of exceptional type G2\mathrm{G}_2. We first show that if Γ\Gamma is a subgroup of Spin(7)\mathrm{Spin}(7), and if each element of Γ\Gamma is conjugate to some element of G2\mathrm{G}_2, then Γ\Gamma itself is conjugate to a subgroup of G2G_2. The analogous statement for SO(7)\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 SO(7)\mathrm{SO}(7) in a very specific way: GL2(Z/3Z)\mathrm{GL}_2(\mathbb{Z}/3\mathbb{Z}), SL2(Z/3Z)\mathrm{SL}_2(\mathbb{Z}/3\mathbb{Z}), Z/4Z×Z/2Z\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}, as well as the nonabelian subgroups of GO2(C)\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 GL7\mathrm{GL}_7 over a totally real number field so that its associated \ell-adic Galois representations can be conjugate into G2(Q)\mathrm{G}_2(\overline{\mathbb{Q}_\ell}). We provide 11 examples over Q\mathbb{Q} which are unramified at all primes.

Cite this article

Gaëtan Chenevier, Subgroups of Spin(7)\mathrm{Spin}(7) or SO(7)\mathrm{SO}(7) with Each Element Conjugate to Some Element of G2\mathrm{G}_2 and Applications to Automorphic Forms. Doc. Math. 24 (2019), pp. 95–161

DOI 10.4171/DM/676