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

Abstract

In one article, the author has defined an L-group associated to a cover of a quasisplit reductive group over a local or global field. In another article, Wee Teck Gan and Fan Gao define (following an unpublished letter of the author) an L-group associated to a cover of a pinned split reductive group over a local or global field. In this short note, we give an isomorphism between these L-groups. In this way, the results and conjectures discussed by Gan and Gao are compatible with those of the author. Both support the same Langlands-type conjectures for covering groups.

Stable conjugacy and epipelagic L-packets for Brylinski-Deligne covers of Sp(2n)

“…For the next result, we recall from [57,Remark 19.8] that π ét 1 (E ǫ ( G)) is realized as lim − →z π ét 1 (E ǫ ( G), z) with respect to unique transition isomorphisms, where the "geometric basepoints" z ranges over objects of E ǫ ( G)( F ). We will employ the concrete description of (5.2) in [56] or [14, §5.2] for pinned split groups, called the second twist in [14]. Take a symplectic basis, which gives rise to a standard F -pinning for G; note that the symplectic bases form a single G(F )-orbit.…”

“…Represent the elements of π ét 1 (E ǫ ( G), z0 ) as (τ, ζ) ∈ T × Z G∨ . To simplify matters, we fix γ ∈ Γ F and look only at the fibers in E 0 , E 1 over q m (γ); it is shown in [56…”