The essentially tame local Langlands correspondence, I

Abstract

Let F F be a non-Archimedean local field (of characteristic 0 0 or p p ) with finite residue field of characteristic p p . An irreducible smooth representation of the Weil group of F F is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension n n is denoted G n e t ( F ) \mathcal {G}^\mathrm {et}_n(F) . The Langlands correspondence induces a bijection of G n e t ( F ) \mathcal {G}^\mathrm {et}_n(F) with a certain set A n e t ( F ) \mathcal {A}^\mathrm {et}_n(F) of irreducible supercuspidal representations of G L n ( F ) \mathrm {GL}_n(F) . We consider the set P n ( F ) P_n(F) of isomorphism classes of certain pairs ( E / F , ξ ) (E/F,\xi ) , called “admissible”, consisting of a tamely ramified field extension E / F E/F of degree n n and a quasicharacter ξ \xi of E × E^\times . There is an obvious bijection of P n ( F ) P_n(F) with G n e t ( F ) \mathcal {G}^\mathrm {et}_n(F) . Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of P n ( F ) P_n(F) with A n e t ( F ) \mathcal {A}^\mathrm {et}_n(F) , generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of G n e t ( F ) \mathcal {G}^\mathrm {et}_n(F) with A n e t ( F ) \mathcal {A}^\mathrm {et}_n(F) . We show that one obtains the Langlands correspondence by composing the map P n ( F ) → A n e t ( F ) P_n(F) \to \mathcal {A}^\mathrm {et}_n(F) with a permutation of P n ( F ) P_n(F) of the form ( E / F , ξ ) ↦ ( E / F , μ ξ ξ ) (E/F,\xi )\mapsto (E/F,\mu _\xi \xi ) , where μ ξ \mu _\xi is a tamely ramified character of E × E^\times depending on ξ \xi . This answers a question of Moy (1986). We calculate the character μ ξ \mu _\xi in the case where E / F E/F is totally ramified of odd degree.

On depth zero L‐packets for classical groups

“…To this end, one seeks to make the correspondence explicit/effective. For GL n , this has been the subject of a series of papers by Bushnell-Henniart [6][7][8]10]; for other groups, work has concentrated on regular depth zero irreducible cuspidal representations [13,24,26] and epipelagic irreducible cuspidal representations [9,18,25,27,46,47], with the most general work by Kaletha [28] on regular cuspidal representations.…”

On the local Langlands and Jacquet–Langlands correspondences

“…The twisting character μ has been computed, at least when E/F is totally ramified [13]. The answer is easy to state only when n is odd; moreover when n is even, the answer does not coincide in general with the recipe conjectured in [49].…”

“…In the tame case Howe parametrized both G o (n) and A o (G n ) in terms of admissible pairs (E/F, θ), where E/F is a degree n extension, θ a character of E × not factorizing through an intermediate norm N F/E F ⊂ E ⊂ E, E = E, and such that if θ restricted to principal units of E does factorize, then E/E is unramified. There is then a canonical map (E/F, θ) → π(E/F, θ) giving a bijection between admissible pairs up to isomorphism and A o (G n ); see [49] for a precise construction, or [12].…”

“…Theorem ( [12]). Let (E/F, θ) be an admissible pair with E/F tamely ramified of degree n. Then there is a unique tamely ramified character μ = μ(E/F, θ) of E × such that π(σ (E/F, θ)) = π(E/F, μθ).…”

Intégrales orbitales tordues sur GL(n, F) et corps locaux proches : applications

“…Ce travail s'inscrit dans une série d'articles [HL1], [HL2], [HL3] dans lesquels les auteurs se proposent d'étendreà la caractéristique non nulle les théories du changement de base et de l'induction automorphe pour G, connues pour F de caractéristique nulle. Par exemple le lemme fondamental démontré ici permettra d'étendreà la caractéristique > 0 le travail de G. Henniart et R. Herb [HH] prouvant l'existence d'une application de relèvement entre représentations de H(= GL(m, E)) et représentations de G. Notons que ces résultats sur l'induction automorphe et le changement de base, sont utilisés dans un travail commun de C. Bushnell et G. Henniart [BH1], [BH2], pour décrire explicitement et en toute caractéristique la correspondance de Langlands entre représentations irréductibles de degré n du groupe de Weil de F et représentations irréductibles cuspidales de G, dans le cas où n est premier a la caractéristique résiduelle de F.…”