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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