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Uniform temperedness of Whittaker integrals for a real reductive group

E. P. van den Ban

Abstract

We study Whittaker vectors (and Jacquet integrals) in the generalized principal series for a real reductive group. A functional equation for them is obtained. This allows to establish uniform estimates for their holomorphic extensions with respect to the continuous induction parameter. Finally, we link the Whittaker vectors to Harish-Chandra's Whittaker integrals for which we then prove uniform tempered estimates. This allows us to establish rapid decay for a class of Fourier transforms on the space of Whittaker Schwartz functions.

Uniform temperedness of Whittaker integrals for a real reductive group

Abstract

We study Whittaker vectors (and Jacquet integrals) in the generalized principal series for a real reductive group. A functional equation for them is obtained. This allows to establish uniform estimates for their holomorphic extensions with respect to the continuous induction parameter. Finally, we link the Whittaker vectors to Harish-Chandra's Whittaker integrals for which we then prove uniform tempered estimates. This allows us to establish rapid decay for a class of Fourier transforms on the space of Whittaker Schwartz functions.
Paper Structure (18 sections, 137 theorems, 685 equations)

This paper contains 18 sections, 137 theorems, 685 equations.

Table of Contents

  1. Whittaker vectors and matrix coefficients
  2. Moderate estimates for Whittaker coefficients
  3. The Whittaker Schwartz space
  4. Sharp estimates for Whittaker coefficients
  5. Proof of Lemma \ref{['l: estimate imp']}: improvement of estimates
  6. Parabolically induced representations
  7. Generalized vectors for induced representations
  8. Whittaker vectors for induced representations
  9. The Whittaker integral
  10. Finite dimensional spherical representations
  11. Projection along infinitesimal characters
  12. The functional equation
  13. Proof of Lemma \ref{['l: generic injectivity of map with j']}
  14. Holomorphy and uniformly moderate estimates
  15. Uniformly tempered estimates
  16. ...and 3 more sections

Key Result

Lemma 1.1

The matrix coefficient map $\lambda \mapsto {\rm wh}_\lambda$ is a bijection where ${\rm Hom}_G$ indicates the space of intertwining continuous linear maps. The inverse of (e: matrix coefficient Whittaker) is given by $T \mapsto {\rm ev}_e \,{\circ}\, T$, where ${\rm ev}_e: C^\infty(G/N_0\!:\! \chi) \rightarrow {{\mathbb C}}$ denotes evaluation at the identity.

Theorems & Definitions (169)

  • Lemma 1.1
  • Lemma 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 159 more