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

[摘要] Let F be a non-Archimedean local field. Let $mathcal{G}_n^{m et}(F)$ 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 $mathcal{A}_n^{m et}(F)$ be the set of equivalence classes of irreducible, essentially tame, supercuspidal representations of GLn(F). The Langlands correspondence induces a canonical bijection $mathcal{L}:mathcal{G}_n^{m et}(F) o mathcal{A}_n^{m et}(F)$. We continue the programme of describing this map in terms of explicit descriptions of the sets $mathcal{G}_n^{m et}(F)$ and $mathcal{A}_n^{m et}(F)$. 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 $E^imes$ subject to certain technical conditions. If Pn(F) is the set of isomorphism classes of admissible pairs of degree n, we have explicit bijections $P_n(F) cong mathcal{G}_n^{m et}(F)$ and $P_n(F) cong mathcal{A}_n^{m et}(F)$. In an earlier paper we showed that, if $sigma in mathcal{G}_n^{m et}(F)$ corresponds to an admissible pair $(E/F,xi)$, then $mathcal{L}(sigma)$ corresponds to the admissible pair $(E/F,muxi)$, for a certain tamely ramified character $mu$ of $E^imes$. In this paper, we determine the character $mu$ when $E/F$ is totally ramified.

[学科分类] 数学（综合）