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- W2418402308 abstract "As is well-known, the compact groups Spin(7) and SO(7) both have a single conjugacy class of compact subgroups of exceptional type G_2. We first show that if H is a subgroup of Spin(7), and if each element of H is conjugate to some element of G_2, then H itself is conjugate to a subgroup of G_2. 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: GL_2(Z/3Z), SL_2(Z/3Z), Z/4Z x Z/2Z, as well as the nonabelian subgroups of GO_2(C) with compact closure, similitude factors group {-1,1}, 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 GL_7 over a totally real number field so that its associated ell-adic Galois representations can be conjugate into G_2(bar{Q_ell})." @default.
- W2418402308 created "2016-06-24" @default.
- W2418402308 creator A5031138502 @default.
- W2418402308 date "2019-01-01" @default.
- W2418402308 modified "2023-10-16" @default.
- W2418402308 title "Subgroups of Spin(7) or SO(7) with each element conjugate to some element of G_2, and applications to automorphic forms" @default.
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