A Remark on a Trace Paley--Wiener Theorem

TL;DR

This paper establishes a version of the trace Paley--Wiener theorem for tempered representations of reductive p-adic groups and applies it to analyze the invariance of Plancherel measures within L-packets.

Contribution

It introduces a new version of the trace Paley--Wiener theorem tailored for p-adic groups and advances understanding of Plancherel measure invariance in the context of L-packets.

Findings

Proved a trace Paley--Wiener theorem for tempered representations.

Applied the theorem to demonstrate invariance of Plancherel measure within L-packets.

Extended Shahidi's work on Plancherel measure invariance.

Abstract

In this paper we prove a version of a trace Paley--Wiener theorem for tempered representations of a reductive $p$--adic group. This is applied to complete certain investigation of Shahidi on the proof that a Plancherel measure is invariant of a $L$--packet of discrete series.