La formule des traces pour les rev\^etements de groupes r\'eductifs connexes. II. Analyse harmonique locale

TL;DR

This paper develops local harmonic analysis tools, including Plancherel formula and character behavior, to support Arthur's invariant trace formula for coverings of reductive groups, leading to results like the density of tempered characters.

Contribution

It introduces new harmonic analysis results for local coverings, crucial for extending Arthur's trace formula to non-linear group coverings.

Findings

Establishment of the Plancherel formula for coverings

Normalization of intertwining operators under Arthur's conditions

Density of tempered characters for coverings

Abstract

We establish some results in local harmonic analysis which are necessary for Arthur's invariant trace formula for coverings of connected reductive groups. More precisely, for local coverings we will study (1) the Plancherel formula and its preparations, (2) the normalization of intertwining operators subject to Arthur's conditions, (3) the local behavior of characters of admissible representations in the nonarchimedean case, and (4) the genuine part of the invariant local trace formula. As a byproduct of the invariant local trace formula, we deduce the density of tempered characters for coverings.