The Explicit Local Langlands Correspondence for $G_2$

Abstract

We develop a general strategy for constructing the explicit Local Langlands Correspondences for $p$-adic reductive groups via reduction to LLC for supercuspidal representations of proper Levi subgroups, using Hecke algebra techniques. As an example of our general strategy, we construct the explicit Local Langlands Correspondence for the exceptional group $G_2$ over a nonarchimedean local field, with explicit $L$-packets and explicit matching between the group and Galois sides. We also give a list of characterizing properties for our LLC. In \cite{G2-stability}, we complete unique characterization using stability property of our $L$-packets. For intermediate series, we build on our previous results on Hecke algebras. For principal series, we improve previous works of Muic etc. and obtain more explicit descriptions on both group and Galois sides. Moreover, we show the existence of non-unipotent \textit{singular} supercuspidal representations of $G_2$, and exhibit them in \textit{mixed} $L$-packets mixing supercuspidal representations with non-supercuspidal ones. Furthermore, our LLC satisfies a list of expected properties, including the compatibility with cuspidal support.