Hilbert and Fr\'echet bundle versions of the Harish-Chandra and Whittaker Plancherel Theorems

TL;DR

This paper provides a new, representation-theoretic proof of the Whittaker Plancherel Theorem using Hilbert and Fréchet bundles, avoiding traditional Eisenstein integral methods.

Contribution

It introduces a bundle-based approach to the Plancherel Theorem, simplifying the proof and extending it to both L^2 and Whittaker cases without relying on Eisenstein integrals.

Findings

Complete proof of the direct integral version of the Whittaker Plancherel Theorem.

Development of a bundle framework over the tempered dual for representation theory.

Simplification of the proof process by bypassing Eisenstein integrals.

Abstract

This paper, in particular, gives a complete proof of the direct integral version of the Whittaker Plancherel Theorem. The main emphasis is on certain Hilbert and Fr\'echet vector bundles over a space that has a submersion onto the tempered dual. This allows for an approach to the Plancherel Theorems (both for L^2 and the Whittaker case) that is representation theoretic, bypasses the need for Harish-Chandra's Eisenstein Integrals and yields a proof the direct integral decompositions without invoking the abstract theory.