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The local Langlands correspondence: The non-Archimedean case. (English) Zbl 0811.11072

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 365-391 (1994).
The local Langlands correspondence is the most important step in the construction of general automorphic \(L\)-functions. The latter are the most promising candidate for motivic \(L\)-functions.
The correspondence should attach to each smooth admissible representation \(\pi\) of a reductive group \(G(F)\) over a local field \(F\) a representation \(\sigma\) of the Weil-Deligne group of \(F\) in a way that certain axioms are satisfied. One then defines the local \(L\)-function of \(\pi\) to be the Artin-Tate \(L\)-function of \(\sigma\) [A. Borel, Proc. Sympos. Pure Math. 33, Part 2, 27-61 (1979; Zbl 0412.10017)].
The paper under consideration presents the state of the art concerning the correspondence in the nonarchimedean case for the group \(GL_ n\). Since all irreducible smooth representations are subquotients of representations induced from supercuspidal representations, the verification of the correspondence conjecture should follow two steps: First prove the conjecture for supercuspidals and then show that it is compatible with induction. The second step has been achieved, the first not in general.
Among other things the paper contains a comprehensive survey of the results of Bernstein and Zelevinski (see references in the text) classifying the irreducible smooth representations in terms of supercuspidal ones. Since \(GL_ n\) is on the one hand the central example, on the other much simpler than the general case, this paper is highly recommendable for those readers who want to understand the \(p\)- adic theory of automorphic forms.
For the entire collection see [Zbl 0788.00054].

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11F85 \(p\)-adic theory, local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields

Citations:

Zbl 0412.10017
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