To an effective local Langlands Corrspondence

Abstract

Let $F$ be a non-Archimedean local field. Let $\Cal W_F$ be the Weil group of $F$ and $\Cal P_F$ the wild inertia subgroup of $\scr W_F$. Let $\hat{\Cal W}_F$ be the set of equivalence classes of irreducible smooth representations of $\Cal W_F$. Let $\Cal A^0_n(F)$ denote the set of equivalence classes of irreducible cuspidal representations of $\roman{GL}_n(F)$ and set $\hat{\roman{GL}}_F = \bigcup_{n\ge1} \Cal A^0_n(F)$. If $\sigma\in \hat{\Cal W}_F$, let $\upr L\sigma \in \hat{\roman{GL}}_F$ be the cuspidal representation matched with $\sigma$ by the Langlands Correspondence. If $\sigma$ is totally wildly ramified, in that its restriction to $\Cal P_F$ is irreducible, we treat $\upr L\sigma$ as known. From that starting point, we construct an explicit bijection $\Bbb N:\hat{\Cal W}_F \to \hat{\roman{GL}}_F$, sending $\sigma$ to $\upr N\sigma$. We compare this "na\"ive correspondence" with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of "internal twisting" of a suitable representation $\pi$ (of $\Cal W_F$ or $\roman{GL}_n(F)$) by tame characters of a tamely ramified field extension of $F$, canonically associated to $\pi$. We show this operation is preserved by the Langlands correspondence.