Chenevier, Gaëtan

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

Doc. Math. 24, 95-161 (2019)
DOI: 10.25537/dm.2019v24.95-161
Communicated by Don Blasius

Summary

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.

Mathematics Subject Classification

20G41, 22C05, 20G15, 11F80, 11R39

Keywords/Phrases

exceptional group \(\mathrm{G}_2\), subgroups of \(\mathrm{SO}(7)\), automorphic forms, Galois representations, Langlands conjectures

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Affiliation

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

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