Formule de Plancherel pour les fonctions de Whittaker sur un groupe réductif p-adique

Abstract

International audienceWe 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.Nous prouvons la formule de Plancherel pour les fonctions de Whittaker sur un groupe réductif p-adique. Les méthodes sont proches de celles de lapreuve de Waldspurger, d’après Harish-Chandra, pour les fonctions lisses sur le groupe.Au delà du résultat, ce travail met en place un cadre qui devrait s’avérer utile pour d’autres formules de Plancherel, notamment pour les espaces symétriques réductifs p-adiques. En particulier, il met en valeur le role des matrices B et de leur propriété d’adjonction

Periods and harmonic analysis on spherical varieties

“…3 While our paper was being written, a complete description of the Whittaker-Plancherel formula was obtained by Delorme [Del13]. Our proof is rather different.…”

An explicit inversion formula for the $p$-adic Whittaker transform on $\text{GL}_n(\mathbb{Q}_p)$

“…In the last section, we use the inversion formula to write integral representations of local L-factors associated to a symmetric square and symmetric cube lifts. We remark that the inversion formula had already been obtained in the setting of p-adic reductive groups by Delorme [4]. However, the approach presented here provides a more explicit presentation of the result in this particular setting and its simple derivation solely relies on complex analysis.…”

A Simple Generalised Plancherel Formula for Compactly Induced Characters

“…Delorme. Delorme [3] has obained an inversion formula for a Whittaker function using the matrix coefficients of certain representations induced parabolically and a version of the Schur Orthogonality relation for such coefficients. Assuming a character ψ to be nondegenerate, Delorme's formula expresses a ψ-Whittaker function in terms of certain transforms of the function itself.…”

“…For a closed subgroup U of G having a continuous unitary character ψ, a function in the space representing the compact induction c-Ind G U ψ also affords a generalised Plancherel formula which provides further information about Ĝ. Such formulae have been worked out in various settings, especially when G is a p-adic group or a Lie group with certain properties Date: October 11, 2018. and U is the unipotent radical of a Borel subgroup of G (notably, the Whittaker-Plancherel formulae by Baruch and Mao [1], Delorme [3] and also Sakellaridis and Venkatesh [10]). In this article, we derive a generalised Plancherel formula for a broad class of groups which generalises such Whittaker-Plancherel type formulae in the pointwise case.…”