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Local constants and the tame Langlands correspondence. (English) Zbl 0597.12019
Let F be a p-adic field, and let $$W_ F$$ be its absolute Weil group. As a part of Langlands’ conjectures, there should be a ”natural” correspondence between the (equivalence classes of) representations $$W_ F\to GL_ n({\mathbb{C}})$$ and the (equivalence classes of) irreducible admissible representations of $$GL_ n(F)$$. Thanks to the work of Berstein-Zelevinsky, the above correspondence is known to exist if one can show the existence of such a correspondence between irreducible n- dimensional representations of $$W_ F$$ and supercuspidal representations of $$GL_ n({\mathbb{C}})$$. The present paper establishes this latter correspondence in the ”tame” case, where $$p| n.$$
There are a number of important intermediate results that the author also obtains. First of all, he gives a parametrization of both sets of representations. For any finite extension E of F, Howe defined a notion of admissible quasicharacters of $$E^{\times}$$, and gave a definition of equivalence of admissible quasicharacters. Say that an admissible quasicharacter is of degree n if the associated field E satisfies $$[E:F]=n$$ (the equivalence relation respects this definition). R. E. Howe [Pac. J. Math. 73, 437-460 (1977; Zbl 0404.22019)] gave an injective map of the set of equivalence classes of admissible quasicharacters of degree n into the set of supercuspidal representations of GL$${}_ n(F)$$. One of the important results of this paper is that this map is bijective. The author proves this by using a theorem of Deligne and Kazhdan giving a bijection between discrete series representations of $$GL_ n$$ and the unitary dual of $$D_ n^{\times}$$, where $$D_ n$$ is a central division algebra of degree $$n^ 2$$ over F. For this part of the paper, and for defining the Langlands correspondence, he needs to compute the ”local constants” - L-factors and $$\epsilon$$-factors - of these representations; he also computes the local constants for the representations of $$W_ F$$. One consequence is that the correspondence of local constants does not define the Langlands correspondence uniquely when $$n\geq 4$$.
Reviewer: L.Corwin

##### MSC:
 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields
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