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Zbl 0666.12013
Henniart, Guy
La conjecture de Langlands locale numérique pour GL(n). (The numerical local Langlands conjecture for GL(n)). (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 21, No. 4, 497-544 (1988). ISSN 0012-9593

For any non-archimedean local field K and any n an injective map is constructed from the set of equivalence classes of irreducible n- dimensional complex representations of the Weil group of K into the set of equivalence classes of irreducible supercuspidal representations of $GL(n,K)$. This map preserves conductors and is compatible with torsion by unramified characters of $K\sp*$. It is proved that all injective maps with those properties are bijective. It is not proved that there is one which preserves $\epsilon$-factors. The surjectivity is derived from a conjecture on numbers of representations formulated by Koch and proved here. \par In characteristic p this conjecture is proved using Laumon's Fourier transform. In characteristic 0 the field K is compared with a local field of characteristic $\ne 0$ with the same residue field. Moreover base change for GL(n) is used. \par In the construction of an injective map base change also plays a role, and L-functions and global arguments are employed.
[J.G.M.Mars]

MSC 2000:: *11S37 Langlands-Weil conjectures, nonabelian class field theory
20G25 Linear algebraic groups over local fields and their integers

Keywords: non-archimedean local field K; complex representations of the Weil group; base change

Cited in: Zbl 0921.11060 Zbl 0675.12008

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