Local converse theorems and Langlands parameters

TL;DR

This paper proposes and proves a local converse theorem for generic Langlands parameters of certain reductive groups over non-Archimedean fields, establishing new results for split groups and classical groups under specific conditions.

Contribution

It introduces a converse theorem for generic Langlands parameters, proves it for split groups, and explores its limitations and variants for classical groups and G2.

Findings

Proves the converse theorem for F-split groups.

Shows the theorem does not apply to SO_{2n}(F) for n≥3.

Establishes a variant for G2(F) and classical groups under certain conjectures.

Abstract

Let $F$ be a non Archimedean local field, and $G$ be the $F$-points of a connected quasi-split reductive group defined over $F$. In this note we propose a converse theorem statement for generic Langlands parameters of $G$ when the Langlands dual group of $G$ is acceptable. We then prove it when $G$ is $F$-split. We also prove that the statement does not apply to $\mathrm{SO}_{2n}(F)$ for certain choices of $F$, as soon as $n\geq 3$.Then we consider a variant which we prove for $G=\mathrm{G}_2(F)$ and all quasi-split classical groups. When $F$ has characteristic zero and assuming the validity of the Gross-Prasad and Rallis conjecture, this latter variant translates via the generic local Langlands correspondence of Jantzen and Liu, into the usual local converse theorems for classical groups expressed in terms of Shahidi's gamma factors.