To an effective local Langlands Corrspondence

TL;DR

This paper constructs an explicit bijection between Weil group representations and GL(n) representations over a non-Archimedean local field, providing an effective description of the local Langlands correspondence, especially in the totally wildly ramified case.

Contribution

It introduces a novel 'internal twisting' operation that is preserved by the Langlands correspondence, enabling a more explicit and effective understanding of the correspondence.

Findings

Constructed an explicit bijection between Weil group and GL(n) representations.

Compared naive and Langlands correspondences to clarify their relationship.

Introduced and validated a new operation of internal twisting preserved by the correspondence.

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.