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On lifting. (English) Zbl 0538.20014
Lie group representations II, Proc. Spec. Year, Univ. Md., College Park 1982-83, Lect. Notes Math. 1041, 209-249 (1984).
[For the entire collection see Zbl 0521.00012.]
Let \(F\) be a global field, \(C_ F\) be the adèle class group of \(F\), \(\varepsilon: C_ F\to C^*\) be a character of order \(n\) and let \(L\supset F\) be the corresponding cyclic extension, \({\mathcal G}=\text{Gal}(L/F).\) Let \(A\) be the adèle ring of \(F\), \(G_ A=\text{GL}_ n(A)\), \({\mathcal F}_ a(G)\) be the set of equivalent classes of irreducible automorphic representations of \(G_ A\) and \({\mathcal F}_ a^{\varepsilon}(G)=\{\pi \in {\mathcal F}_ a(G);\;\pi \otimes \varepsilon \cong \pi \}\) where one considers \(\varepsilon\) as a character of \(G_ A\). For any place \({\mathfrak p}\) of \(F\) we denote by \(F_{{\mathfrak p}}\) the completion of \(F\) at \({\mathfrak p}\) and by \(\varepsilon_{{\mathfrak p}}: F^*_{{\mathfrak p}}\to C^*\) the restriction of \(\varepsilon\) to \(F^*_{{\mathfrak p}}\). Let \({\mathfrak p}_ 1\), \({\mathfrak p}_ 2\) be two non-archimedean places such that \(\varepsilon_{{\mathfrak p}_ 1}\), \(\varepsilon_{{\mathfrak p}_ 2}\) are primitive characters of order \(n\),
\(\tilde {\mathcal F}_ a(G)=\{\pi \in {\mathcal F}^{\varepsilon}_ a(G)\); local components of \(\pi\) at \({\mathfrak p}_ 1\) and \({\mathfrak p}_ 2\) are cuspidal} and
\({\hat C}^ 0_ L=\{\chi \in \hat C_ L=\operatorname{Hom}(C_ L,C^*)\); \(\chi^{\sigma}\neq \chi_{{\mathfrak p}_ 1}\), \(\chi^{\sigma}\neq \chi_{{\mathfrak p}_ 2}\) for any \(\sigma\in G-\{e\}\), where \(\chi_{{\mathfrak p}_ 1},\chi_{{\mathfrak p}_ 2}\) are local components of \(\chi\) at \({\mathfrak p}_ 1\) and \({\mathfrak p}_ 2\}\).
The author first proves that there exists a natural one-to-one correspondence between \(\tilde {\mathcal F}^{\varepsilon}_ a(G)\) and \({\mathcal G}\)-orbits on \({\hat C}^ 0_ L\). Using Arthur’s results with this theorem, one can show that there exists a “natural” one-to-one correspondence between \({\mathcal F}^{\varepsilon}_ a(G)\) and \({\mathcal G}\)-orbits on \({\hat C}_ L\). Also, the author proves the following local variant of the theorem. Let \(F\) be a local non-archimedean field, \(\varepsilon:F^*\to C^*\) be a character of order \(n\). \(G=\text{GL}_ n(F)\). We denote by the same letter \(\varepsilon\) the character of \(G\) given by \(\varepsilon(g)=\varepsilon(\det g),\) \(g\in G\). Let \({\mathcal F}(G)\) be the set of equivalent classes of smooth irreducible representations of \(G\) and \({\mathcal F}^{\varepsilon}(G)=\{\pi \in {\mathcal F}(G);\;\pi \otimes \varepsilon \simeq \pi \}\). Let \(L\supset F\) be the cyclic extension which corresponds to \(\varepsilon\) by the local class field theory and \(G=\text{Gal}(L/F)\,(\cong \mathbb Z/n\mathbb Z)\). Then there exists a natural one-to-one correspondence between \({\mathcal F}^{\varepsilon}(G)\) and \({\mathcal G}\)-orbits on \(\operatorname{Hom}(L^*,G^*)\).
Reviewer: E. Abe

MSC:
20G05 Representation theory for linear algebraic groups
20G35 Linear algebraic groups over adèles and other rings and schemes
11R39 Langlands-Weil conjectures, nonabelian class field theory
20G25 Linear algebraic groups over local fields and their integers
20G30 Linear algebraic groups over global fields and their integers
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields