Compatibility of semisimple local Langlands parameters with parahoric Satake parameters

TL;DR

This paper establishes the uniqueness of the correspondence between parahoric-spherical representations and semisimple local Langlands parameters, confirming a conjecture about their compatibility with Satake parameters using a formal proof approach.

Contribution

It proves the uniqueness of the correspondence satisfying natural properties and confirms the compatibility conjecture for semisimple local Langlands parameters and parahoric Satake parameters.

Findings

Uniqueness of the correspondence under natural properties

Confirmation of the compatibility conjecture

Semisimple parameters by Fargues and Scholze are the unique candidates

Abstract

In this paper, we prove that there is at most one correspondence between parahoric-spherical representations and semisimple local Langlands parameters which satisfies certain natural properties. Our proof of this uniqueness statement is very formal. In particular, the semisimple local Langlands parameters constructed by Fargues and Scholze yield the unique candidate for such representations. As a corollary, we prove a conjecture which posits the compatibility of semisimple local Langlands parameters with parahoric Satake parameters.