On unipotent representations of ramified $p$-adic groups

TL;DR

This paper establishes a local Langlands correspondence for unipotent representations of ramified p-adic groups, extending previous work to more general reductive groups and confirming key conjectures about formal degrees.

Contribution

It generalizes existing theories to ramified groups and interprets results through rigid inner twists, advancing understanding of unipotent representations.

Findings

Established a local Langlands correspondence for ramified groups.

Confirmed conjectures on formal degrees for these representations.

Extended previous work to non-split reductive groups.

Abstract

Let G be any connected reductive group over a non-archimedean local field. We analyse the unipotent representations of G, in particular in the cases where G is ramified. We establish a local Langlands correspondence for this class of representations, and we show that it satisfies all the desiderata of Borel as well as the conjecture of Hiraga, Ichino and Ikeda about formal degrees. This generalizes work of Lusztig and of Feng, Opdam and the author, to reductive groups that do not necessarily split over an unramified extension of the ground field. We also interpret our results in terms of rigid inner twists of G.