Covers of extended pure inner forms and Weissman’s obstruction

Covering groups are topological central extensions of G(F) by a finite abelian group, where F is a finite extension of Q_p and G is a reductive group over F. Weissman observed that the local Langlands correspondence for covering groups fails to be surjective in general. Our result shows that by considering covers of all extended pure inner forms of G, the surjectivity issue can be resolved in the case of covering tori.

In the first part of the talk I will present Brylinski-Deligne’s construction of
covering groups, explain the notion of Weissman’s obstruction for surjectivity and the statement of the main theorem. In the second half of the talk, I will discuss the construction of covers of extended pure inner forms using the language of étale metaplectic covers and the proof strategy. This talk is based on joint work with Yifei Zhao.