As is well-known, the compact groups and both have a single conjugacy class of compact subgroups of exceptional type . We first show that if is a subgroup of , and if each element of is conjugate to some element of , then itself is conjugate to a subgroup of . The analogous statement for 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 in a very specific way: , , , as well as the nonabelian subgroups of with compact closure, similitude factors group , 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 over a totally real number field so that its associated -adic Galois representations can be conjugate into . We provide 11 examples over which are unramified at all primes.
Cite this article
Gaëtan Chenevier, Subgroups of or with Each Element Conjugate to Some Element of and Applications to Automorphic Forms. Doc. Math. 24 (2019), pp. 95–161DOI 10.4171/DM/676