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

TL;DR

This paper establishes an isomorphism between two independently defined L-groups for covers of split reductive groups, ensuring compatibility of their associated Langlands conjectures.

Contribution

It provides a formal proof that the L-groups defined by different authors are isomorphic, unifying their frameworks for covering groups.

Findings

The L-groups are isomorphic for covers of split reductive groups.

Supports the compatibility of Langlands conjectures across different L-group definitions.

Bridges the gap between different approaches to L-groups in the literature.

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 In another article, Wee Teck Gan and Fan Gao define (following an unpublished letter of the author) an L-group associated to a cover of a pinned split reductive group over a local or global field.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.