Local constants and the tame Langlands correspondence.

*(English)*Zbl 0597.12019Let 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 \(n\geq 4\).

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 |