What are the extended pure inner forms of a cover?

TL;DR

This paper explores the classification of extended pure inner forms of covers within the local Langlands framework, extending Kottwitz's ideas to Brylinski-Deligne covers and discussing implications for L-packets.

Contribution

It generalizes the concept of extended pure inner forms to a broad class of covers, connecting with Weissman's observations on L-packets for these covers.

Findings

Extended pure inner forms are classified for Brylinski-Deligne covers.

Relation established between inner forms and the emptiness of L-packets.

Provides a framework linking Kottwitz's philosophy to covers in the Langlands program.

Abstract

Kottwitz suggested to study all extended pure inner forms together in the local Langlands correspondence for linear reductive groups. We extend this philosophy to a large class of covers, including those defined by Brylinski and Deligne, and explain its relation with Weissman's observation that L-packets for covers are sometimes empty.