A Simple Generalised Plancherel Formula for Compactly Induced Characters

TL;DR

This paper introduces a simplified, generalized Plancherel formula applicable to various locally compact groups, unifying and extending existing formulas for specific groups like p-adic and metaplectic groups.

Contribution

It presents a new, simplified generalized Plancherel formula for a broad class of groups, unifying previous specialized formulas and extending applicability to metaplectic and other groups.

Findings

Unified Plancherel formula for unimodular groups

Specializes to known formulas for p-adic groups

Applicable to metaplectic and distinguished representations

Abstract

The aim of this article is to present a simple generalized Plancherel formula for a locally compact unimodular topological group G of type I. This formula specializes to the Whittaker-Plancherel formula for a split reductive p-adic group of Sakellaridis-Venkatesh and differs from that of a quasi-split p-adic group due to Delorme. Furthermore, it also applies to certain metaplectic groups and other interesting situations where the local theory of distinguished representations has been studied.