Let $G$ be a connected reductive group over the non-archime-dean local field $F$ and let $pi$ be a supercuspidal representation of $G(F)$. The local Langlands conjecture posits that to such a $pi$ can be attached a parameter $L(pi)$, which is an equivalence class of homomorphisms from the Weil-Deligne group with values in the Langlands $L$-group ${}^LG$ over an appropriate algebraically closed field $C$ of characteristic $0$. When $F$ is of positive characteristic $p$ then Genestier and Lafforgue have defined a parameter, $L^{ss}(pi)$, which is a homomorphism $W_F ra {}^LG(C)$ that is {it semisimple} in the sense that, if the image of $L^{ss}(pi)$, intersected with the Langlands dual group $hat{G}(C)$, is contained in a parabolic subgroup $P subset hat{G}(C)$, then it is contained in a Levi subgroup of $P$. If the Frobenius eigenvalues of $L^{ss}(pi)$ are pure in an appropriate sense, then the local Langlands conjecture asserts that the image of $L^{ss}(pi)$ is in fact {it irreducible} -- its image is contained in no proper parabolic $P$. In particular, unless $G = GL(1)$, $L^{ss}(pi)$ is ramified: it is non-trivial on the inertia subgroup $I_F subset W_F$. In this paper we prove, at least when $G$ is split and semisimple, that this is the case provided $pi$ can be obtained as the induction of a representation of a compact open subgroup $U subset G(F)$, and provided the constant field of $F$ is of order greater than $3$. Conjecturally every $pi$ is compactly induced in this sense, and the property was recently proved by Fintzen to be true as long as $p$ does not divide the order of the Weyl group of $G$. The proof is an adaptation of the globalization method of cite{GLo} when the base curve is $PP^1$, and a simple application of Delignes Weil II.
In this joint introduction to an Asterisque volume, we give a short discussion of the historical developments in the study of nonlinear covering groups, touching on their structure theory, representation theory and the theory of automorphic forms. This serves as a historical motivation and sets the scene for the papers in the volume. Our discussion is necessarily subjective and will undoubtedly leave out the contributions of many authors, to whom we apologize in earnest.
