L-groups and parameters for covering groups

Abstract

We incorporate covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. We work with all covers that arise from extensions of quasisplit reductive groups by K 2the class studied by Brylinski and Deligne. We use this L-group to parameterize genuine irreducible representations in many contexts, including covers of split tori, unramified representations, and discrete series for double covers of semisimple groups over R. An appendix surveys torsors and gerbes on the étale site, as they are used in the construction of the L-group.

A comparison of L-groups for covers of split reductive groups

“…Fix an injective character ǫ : µ n (F ) ֒→ C × . From this data, the constructions of [Wei15] and [GG14] both yield an L-group of G via a Baer sum of two extensions. In both papers, an extension Z∨ ֒→ E 2 ։ Gal F , following an unpublished letter (June, 2012) from the author to Deligne.…”

“…The following notions of covering groups and their dual groups match those in [Wei15]. Let G = (G ′ , n) be a degree n cover of G over F ; in particular, #µ n (F ) = n. Here G ′ is a central extension of G by K 2 in the sense of [B-D], and write (Q, D, f ) for the three Brylinski-Deligne invariants of G ′ .…”

“…In both papers, an extension Z∨ ֒→ E 2 ։ Gal F , following an unpublished letter (June, 2012) from the author to Deligne. In [Wei15], the second twist is the fundamental group of a gerbe, denoted π ét 1 (E ǫ ( G), s). In this article s = Spec( F ), and so we write π ét 1 (E ǫ ( G), F ) instead.…”

“…Both papers proceed by taking the Baer sum of these two extensions, E = E 1 ∔ E 2 , to form an extension Z∨ ֒→ E ։ Gal F . The extension E is denoted L Z in [Wei15,§5.4]. Then, one pushes out the extension E via Z∨ ֒→ G∨ , to define the L-group…”

Extensions by $\mathbf K_2$ and factorization line bundles

“…The work of J.-L. Brylinski and P. Deligne [BD01] shows that a large class of such covering groups can be obtained from certain K-theoretic data, which we will refer to as Brylinski-Deligne data. By the work of M. Weissman [We15], their L-groups can be defined and used to parametrize irreducible representations in many contexts. 0.1.3.…”