On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups

TL;DR

This paper explores the connection between two nonabelian Fourier transforms related to unipotent elliptic representations of split p-adic groups, illustrating the relationship with an example involving the G_2 group.

Contribution

It establishes a link between Fourier transforms defined via Langlands parameters and those via pseudocoefficients, providing new insights into the harmonic analysis of p-adic groups.

Findings

Relation between two nonabelian Fourier transforms demonstrated

Explicit example for p-adic group of type G_2 provided

Enhances understanding of harmonic analysis in p-adic representation theory

Abstract

In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig's nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G_2.