Langlands Classification for L-Parameters

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 $π$ of $G$ in terms of triples $(P,σ,ν)$ where $P\subset G$ is a standard $F$-parabolic subgroup, $σ$ is an irreducible tempered representation of the standard Levi-group $M_P$ and $ν\in \Bbb{R}\otimes X^*(M_P)$ is regular with respect to $P.$ Now we consider Langlands' L-parameters $[ϕ]$ which conjecturally will serve as a system of parameters for the representations $π$ and which are (roughly speaking) equivalence classes of representations $ϕ$ of the absolute Galois group $Γ=\text{Gal}(\overline{F}|F)$ with image in Langlands' L-group $\,^LG$, and we classify the possible $[ϕ]$ in terms of triples $(P,[\,^tϕ],ν)$ where the data $(P,ν)$ are the same as in the Langlands classification of representations and where $[\,^tϕ]$ is a tempered L-parameter of $M_P.$