The essentially tame local Langlands correspondence, I

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 $\boldsymbol{\mathscr{G}}^{et}_n(F)$. The Langlands correspondence induces a bijection of $\boldsymbol{\mathscr{G}}^{et}_n(F)$ with a certain set $\boldsymbol{\mathscr{A}}^{et}_n(F)$ of irreducible supercuspidal representations of ${GL}_n(F)$. 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 $\boldsymbol{\mathscr{G}}^{et}_n(F)$. Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of $P_n(F)$ with $\boldsymbol{\mathscr{A}}^{et}_n(F)$, generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of $\boldsymbol{\mathscr{G}}^{et}_n(F)$ with $\boldsymbol{\mathscr{A}}^{et}_n(F)$. We show that one obtains the Langlands correspondence by composing the map $P_n(F) \to \boldsymbol{\mathscr{A}}^{et}_n(F)$ 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.