[Paper Review] L-groups and parameters for covering groups

Martin H. Weissman|arXiv (Cornell University)|Jul 3, 2015

Advanced Algebra and Geometry4 citations

TL;DR

This paper extends the Langlands program to covering groups by defining an L-group for degree-n covers of quasisplit reductive groups over local and global fields, using invariants Q, D, and f from Brylinski-Deligne extensions. It constructs a non-split L-group via the metaGalois group and a gerbe on the étale site, enabling parameterization of genuine irreducible representations, including unramified and discrete series types.

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 $\mathbf{K}_2$ -- the 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 $\mathbb R$. An appendix surveys torsors and gerbes on the étale site, as they are used in the construction of the L-group.

Motivation & Objective

• . The paper aims to incorporate covers of quasisplit reductive groups into the Langlands program.
• It addresses the lack of a systematic L-group construction for covering groups, especially those arising from K2-extensions.
• The objective is to define an L-group that parameterizes genuine irreducible representations in local, global, and arithmetic contexts.
• It seeks to unify representation-theoretic constructions with Galois-theoretic and cohomological tools via gerbes and torsors.
• The work provides a framework for L-functions and Weil parameters in the metaplectic setting.

Proposed method

• . Constructs the L-group L˜G as an extension of GalF by ˜G∨, where ˜G∨ is a pinned complex reductive group.
• Uses three invariants from Brylinski-Deligne theory: a Galois-invariant quadratic form Q, a central extension D of the cocharacter lattice, and a higher invariant f.
• Introduces the metaGalois group as a canonical 2-cocycle extension µ2 → gGalF → GalF via the Hilbert symbol.
• Defines a gerbe Eǫ(˜G) on the étale site F´et, which acts as the second twist in the L-group construction.
• Combines the metaGalois extension and the fundamental group of the gerbe via Baer sum to form the L-group extension.
• Applies the L-group to parameterize ǫ-genuine representations via Weil parameters into L˜G, with orbits under ˜G∨-action.

Experimental results

Research questions

• RQ1 . How can the Langlands program be extended to covering groups arising from K2-extensions of quasisplit reductive groups?
• RQ2 . What is the correct generalization of the L-group for such covers, and how does it differ from the classical Langlands L-group?
• RQ3 . How can genuine irreducible representations be parameterized using this extended L-group?
• RQ4 . What role do gerbes and torsors play in constructing the L-group for covering groups?
• RQ5 . How does the L-group behave under restriction to Levi subgroups and passage between local and global fields?

Key findings

• . The L-group L˜G is constructed as an extension ˜G∨ → L˜G → GalF using the metaGalois group and the fundamental group of a gerbe on F´et.
• . The construction is functorial for 'well-aligned' homomorphisms and respects Levi subgroups and global-local restrictions.
• . The L-group parameterizes ǫ-genuine irreducible representations via ˜G∨-orbits of Weil parameters WF → L˜G.
• . For split tori, the parameterization via Weil parameters agrees with Deligne’s construction and recovers the classical local Langlands correspondence in the metaplectic case.
• . The L-group construction agrees with the E1 + E2 method of [GG14] in the case of split tori, though not yet verified for split reductive groups.
• . The paper provides a systematic framework for L-functions and automorphic L-functions in the metaplectic setting, particularly for unramified and discrete series representations.