Towards the Jacquet Conjecture on the Local Converse Problem for $p$-adic $\mathrm{GL}_n$

TL;DR

This paper addresses the Local Converse Problem for p-adic GL(n), proposing a new approach to prove the Jacquet conjecture that gamma factors determine representations, and verifies it in specific cases like level zero representations.

Contribution

It formulates a general approach to prove the Jacquet conjecture and verifies it in several cases, advancing understanding of gamma factors' role in representation classification.

Findings

The Jacquet conjecture is proved under an assumption.

The approach is verified for level zero representations.

The method advances the classification of representations via gamma factors.

Abstract

The Local Converse Problem is to determine how the family of the local gamma factors $\gamma(s,\pi\times\tau,\psi)$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.