Rigid inner forms vs isocrystals

TL;DR

This paper compares two formulations of the refined local Langlands correspondence for reductive groups over p-adic fields, establishing their equivalence under broad conditions and analyzing their dependence on cohomological choices.

Contribution

It demonstrates the equivalence of two different statements of the refined local Langlands correspondence and explores their dependence on cohomological parameters.

Findings

Equivalence of the two formulations if valid for all connected reductive groups.

Dependence of the second statement on the choice of element in H^1(u -> W,Z -> G).

Provides insights into the structure of the refined local Langlands correspondence.

Abstract

We compare two statements of the refined local Langlands correspondence for connected reductive groups defined over a p-adic field -- one involving Kottwitz's set B(G) of isocrystals with additional structure, and one involving the cohomology set H^1(u -> W,Z -> G) introduced in arXiv:1304.3292. We show that if either statement is valid for all connected reductive groups, then so is the other. We also discuss how the second statement depends on the choice of element of H^1(u -> W,Z -> G).