The L-group of a covering group

TL;DR

This paper defines an L-group for nonlinear covers of quasisplit reductive groups, extending the Langlands program to include these covers and ensuring functoriality and consistency with prior work.

Contribution

It introduces a new L-group construction for covering groups, incorporating extensions by K_2 and functorial properties, advancing the Langlands program for nonlinear covers.

Findings

L-group is an extension of Galois group by a complex reductive group

Construction depends on extension data, degree, character, and field closure

Ensures functoriality and compatibility with previous theories

Abstract

We incorporate nonlinear covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. This L-group is an extension of the absolute Galois group of a local or global field $F$ by a complex reductive group. The L-group depends on an extension of a quasisplit reductive $F$-group by $\mathbf{K}_2$, a positive integer $n$ (the degree of the cover), an injective character $\epsilon \colon \mu_n \rightarrow {\mathbb C}^\times$, and a separable closure of $F$. Our L-group is consistent with previous work on covering groups, and its construction is contravariantly functorial for certain well-aligned homomorphisms. An appendix surveys torsors and gerbes on the \'etale site, as they are used in a crucial step in the construction.