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

Henniart, Guy

Invent. Math. 139, No. 2, 439-455 (2000).

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.

Reviewer: Olaf Ninnemann (Berlin)

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

Citations:

Zbl 1036.11027