Covers of reductive groups and functoriality


Langlands’ reciprocity conjecture parameterizes smooth irreducible representations of a reductive group $G$ over a local field, or automorphic representations of a reductive group $G$ over a global field, in terms of the $L$-group of $G$, which is a split extension of the dual group of $G$ and the Galois group. Langlands’ functoriality conjecture predicts relationship between such representations given a homomorphism between the $L$-groups. In practice, one is often confronted with extensions of the dual group by the $L$-group that are not split. We will explore this phenomenon based on the following construction. 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. We will present one concrete application: a characterization of the local Langlands correspondence for semi-simple discrete $L$-parameters, that is uniform for all local fields.