Covers of reductive groups and functoriality

Abstract: To a connected reductive group $G$ over a local field $F$ we define a compact topological group $\tilde\pi_1(G)$ and an extension $G(F)_\infty$ of $G(F)$ by $\tilde\pi_1(G)$. From any character $x$ of $\tilde\pi_1(G)$ of order $n$ we obtain an $n$-fold cover $G(F)_x$ of the topological group $G(F)$. We also define an $L$-group for $G(F)_x$, which is a usually non-split extension of the Galois group by the dual group of G, and deduce from the linear case a refined local Langlands correspondence between genuine representations of $G(F)_x$ and $L$-parameters valued in this $L$-group.

This construction is motivated by Langlands functoriality. We show that a subgroup of the $L$-group of $G$ of a certain kind naturally lead to a smaller quasi-split group $H$ and a double cover of $H(F)$. Genuine representations of this double cover are expected to be in functorial relationship with representations of $G(F)$. We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal $L$-parameters when $p$ is sufficiently large, and one to the theory of endoscopy.

number theory

Audience: researchers in the topic

Harvard number theory seminar