## La conjecture de Langlands locale pour GL(3). (The local Langlands conjecture for GL(3)).(French)Zbl 0577.12011

In this paper the local Langlands conjecture for GL(3) is proved: Let F be a non-Archimedean local field. There is a unique map from the set of equivalence classes of $$\Phi$$-semisimple complex representations of degree 3 of the Weil-Deligne group of F to the set of equivalence classes of irreducible admissible representations of GL(3,F) preserving L- and $$\epsilon$$-factors. The proof of this theorem and of the properties of the map can be reduced to the proof of the existence, uniqueness and bijectivity of a map from the subset of irreducible representations of the Weil group to the subset of supercuspidal representations of GL(3,F), preserving $$\epsilon$$-factors and commuting with torsion by characters of $$F^{\times}.$$
By known global results the existence of the supercuspidal representation corresponding to a given irreducible representation of the Weil group is ensured in all cases except when F is an extension of $${\mathbb{Q}}_ 3$$ and the representation is primitive. This difficult case is treated by the author using a tame base change. It is necessary to have an explicit description of all irreducible supercuspidal representations of GL(3,F); for this, the technique of Carayol is used, and in order to prove that the description is complete, the author uses a bijection between the irreducible supercuspidal representation of GL(3,F) and the irreducible admissible representations of the multiplicative group of a central division algebra of dimension 9 over F. A proof of the local Langlands conjecture for GL(2) is also indicated, and used in the proof for GL(3).
Reviewer: J.G.M.Mars

### MSC:

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