Supercuspidal unipotent representations: L-packets and formal degrees

TL;DR

This paper establishes a local Langlands correspondence for supercuspidal unipotent representations of certain p-adic groups, linking them to L-parameters and verifying conjectures about their formal degrees.

Contribution

It constructs a bijection between supercuspidal unipotent representations and cuspidal enhanced L-parameters, and confirms the Hiraga-Ichino-Ikeda conjecture on formal degrees.

Findings

Bijection characterized by equivariance properties

Counting of L-packets via affine Dynkin diagram data

Verification of the Hiraga-Ichino-Ikeda conjecture

Abstract

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $\gamma$-factors of L-parameters. This can be regarded as a local Langlands correspondence for all supercuspidal unipotent representations. We count the ensueing L-packets, in terms of data from the affine Dynkin diagram of G. Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.