Langlands Classification for L-Parameters

TL;DR

This paper extends the Langlands classification from irreducible admissible representations of reductive groups over non-archimedean local fields to their L-parameters, classifying these parameters in terms of triples involving parabolic subgroups, tempered L-parameters, and regular elements.

Contribution

It provides a classification of Langlands' L-parameters for reductive groups over non-archimedean fields, paralleling the classification of representations.

Findings

Classifies L-parameters using triples similar to the representation classification.

Establishes a correspondence between L-parameters and triples involving parabolic subgroups and tempered parameters.

Provides a framework for understanding L-parameters in the context of the Langlands program.

Abstract

Let $F$ be a non-archimedean local field and $G={\bf{G}}(F)$ the group of $F$-rational points of a connected reductive $F$-group. Then we have the Langlands classification of complex irreducible admissible representations $\pi$ of $G$ in terms of triples $(P,\sigma,\nu)$ where $P\subset G$ is a standard $F$-parabolic subgroup, $\sigma$ is an irreducible tempered representation of the standard Levi-group $M_P$ and $\nu \in \Bbb{R}\otimes X^*(M_P)$ is regular with respect to $P.$ Now we consider Langlands' L-parameters $[\phi]$ which conjecturally will serve as a system of parameters for the representations $\pi$ and which are (roughly speaking) equivalence classes of representations $\phi$ of the absolute Galois group $\Gamma=\text{Gal}(\overline{F}|F)$ with image in Langlands' L-group $\,^LG$, and we classify the possible $[\phi]$ in terms of triples $(P,[\,^t\phi],\nu)$ where the data $(P,\nu)$ are the same as in the Langlands classification of representations and where $[\,^t\phi]$ is a tempered L-parameter of $M_P.$