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Hilbert and Fréchet bundle versions of the Harish-Chandra and Whittaker Plancherel Theorems

Nolan R. Wallach

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 Fréchet 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.

Hilbert and Fréchet bundle versions of the Harish-Chandra and Whittaker Plancherel Theorems

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 Fréchet 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.

Paper Structure

This paper contains 9 sections, 42 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The Plancherel Space of $G$
  4. The Jacquet integrals and the Whittaker Schwartz space
  5. A technical lemma and a key result
  6. The Whittaker Plancherel space
  7. The Hilbert bundle version of the Whittaker Plancherel Theorem
  8. The direct integral version of the Plancherel Theorem
  9. Some simple results

Key Result

Proposition 1

The following are equivalent for cuspidal parabolic subgroups $P$ and $Q$. 1. $P$ and $Q$ are associate. 2. The Lie algebra of a compact Cartan subgroup of $M_{P}$ is conjugate to a Lie algebra of compact subgroup of $M_{Q}$ 3. $A_{P}$ and $A_{Q}$ are conjugate relative to $G$. 4. The Cartan subgrou

Theorems & Definitions (42)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 32 more