Langlands Classification for L-Parameters

Let $F$ be a non-archimedean local field and $G=\mathbf{G}(F)$ 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 $\nu \in \Bbb{R}\otimes X^*(M_P)$isregularwithrespectto$P.$ Now we consider Langlands’ L-parameters $[\varphi ]$ which conjecturally will serve as a system of parameters for the representations $\pi$ and which are (roughly speaking) equivalence classes of representations $\varphi$ of the absolute Galois group $\mathrm{\Gamma }=\text{Gal}(\bar{F}\mathrm{\mid }F)$ with image in Langlands’ L-group ${}^{L}G$, and we classify the possible $[\varphi ]$ in terms of triples $(P,[{}^t\varphi ],\nu )$ where the data $(P,\nu )$ are the same as in the Langlands classification of representations and where $[{}^t\varphi ]$ is a tempered L-parameter of $M_P\mathrm{.}$