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A simple proof of Langlands conjectures for $$\text{GL}_n$$ on a $$p$$-adic field. (Une preuve simple des conjectures de Langlands pour $$\text{GL}_n$$ sur un corps $$p$$-adique.) (French) Zbl 1048.11092
Let $$F$$ be a finite extension of $$\mathbb Q_p$$. For each integer $$n\geq 1$$, the author constructs a bijection from the set $$\mathcal G_F^0(n)$$ of isomorphism classes of irreducible degree $$n$$ representations of the (absolute) Weil group of $$F$$, onto the set $$\mathcal A_F^0(n)$$ of isomorphism classes of smooth irreducible supercuspidal representations of $$\text{GL}_n(F)$$. Those bijections preserve epsilon factors for pairs and hence the author obtains a proof of the Langlands conjectures for $$\text{GL}_n$$ over $$F$$, which is more direct than that of M. Harris and R. Taylor in [The geometry and cohomology of some simple Shimura varieties. (Annals of Math. Studies 151, Princeton Univ. Press), (2001; Zbl 1036.11027)]. The author’s approach is global and analogous to the derivation of local class field theory from global class field theory. He starts with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and uses a trick of Harris, who constructs non-Galois automorphic induction in certain cases.

##### MSC:
 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields
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