The essentially tame local Langlands correspondence, I

Abstract:

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