The essentially tame local Langlands correspondence, II: totally ramified representations

Abstract

Let F be a non-Archimedean local field. Let $\mathcal{G}_n^{\rm et}(F)$ be the set of equivalence classes of irreducible, n-dimensional representations of the Weil group $\mathcal{W}_F$ of F which are essentially tame. Let $\mathcal{A}_n^{\rm et}(F)$ be the set of equivalence classes of irreducible, essentially tame, supercuspidal representations of GL_n(F). The Langlands correspondence induces a canonical bijection $\mathcal{L}:\mathcal{G}_n^{\rm et}(F) \to \mathcal{A}_n^{\rm et}(F)$. We continue the programme of describing this map in terms of explicit descriptions of the sets $\mathcal{G}_n^{\rm et}(F)$ and $\mathcal{A}_n^{\rm et}(F)$. These descriptions are in terms of admissible pairs $(E/F, \xi)$, consisting of a tamely ramified field extension $E/F$ of degree n and a quasicharacter $\xi$ of $E^\times$ subject to certain technical conditions. If P_n(F) is the set of isomorphism classes of admissible pairs of degree n, we have explicit bijections $P_n(F) \cong \mathcal{G}_n^{\rm et}(F)$ and $P_n(F) \cong \mathcal{A}_n^{\rm et}(F)$. In an earlier paper we showed that, if $\sigma \in \mathcal{G}_n^{\rm et}(F)$ corresponds to an admissible pair $(E/F,\xi)$, then $\mathcal{L}(\sigma)$ corresponds to the admissible pair $(E/F,\mu\xi)$, for a certain tamely ramified character $\mu$ of $E^\times$. In this paper, we determine the character $\mu$ when $E/F$ is totally ramified.