Formule de Plancherel pour les fonctions de Whittaker sur un groupe r\'eductif $p$-adique

TL;DR

This paper establishes the Plancherel formula for Whittaker functions on reductive p-adic groups, building on prior work and simplifying the Fourier transform proof using Bernstein's results.

Contribution

It provides the Plancherel formula for Whittaker functions on reductive p-adic groups, extending previous results and simplifying key aspects of the proof.

Findings

Proved the Plancherel formula for Whittaker functions on p-adic groups.

Connected the proof to Waldspurger's approach for Harish-Chandra's formula.

Simplified the Fourier transform proof using Bernstein's theorem.

Abstract

We prove the Plancherel formula for Whittaker functions on a reductive p-adic group. This a sequel to our work on Paley-Wiener theorem. Our proof is close to the proof written by Waldspurger of the Harish-Chandra Plancherel formula for smooth functions on the group and use many of his results. One simplification is the easy proof of the Fourier transfom, which follows from a result of Joseph Bernstein.