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Maass-Selberg relations for Whittaker functions on a real reductive group

Erik P. van den Ban

TL;DR

This work delivers a complete proof of the Maass-Selberg relations for Whittaker integrals on real reductive groups by reducing the general case to Harish-Chandra's basic setting through standard intertwining operators and Weyl-group symmetries. It introduces generalized Whittaker vectors $j(Q, \sigma, \nu)$ for arbitrary parabolics, defines the $B$- and $C$-functions via intertwiners and constant terms, and proves their Maass-Selberg relations; these results imply regularity of normalized Whittaker integrals and underpin the continuity of normalized Fourier and Wave packet transforms on Harish-Chandra type Schwartz spaces. A key methodological theme is reducing to the maximal-parabolic (adjacent/opposite) case using a root-based α-reduction and then transferring MS properties from the basic setting to the general one. The analysis culminates in Harish-Chandra–style radial Casimir calculations and a boundary-term Gauss-divergence argument that control asymptotics and establish the functional equations and normalization required for the Fourier/Wave-packet theory.

Abstract

We give a complete proof of the Maass-Selberg relations for Whittaker integrals on a real reductive group. These relations were asserted In unpublished work of Harish-Chandra, and proven in the basic setting of maximal parabolic parabolic subgroups. Our proof is a reduction to the mentioned basic setting. It makes use of the action of standard intertwining operators on Whittaker distribution vectors in the generalized principal series of representations. The Maass-Selberg relations imply regularity of normalized Whittaker integrals. This is crucial for decent behavior of Fourier and wave packet transforms on the level of spherical Schwartz spaces.

Maass-Selberg relations for Whittaker functions on a real reductive group

TL;DR

This work delivers a complete proof of the Maass-Selberg relations for Whittaker integrals on real reductive groups by reducing the general case to Harish-Chandra's basic setting through standard intertwining operators and Weyl-group symmetries. It introduces generalized Whittaker vectors for arbitrary parabolics, defines the - and -functions via intertwiners and constant terms, and proves their Maass-Selberg relations; these results imply regularity of normalized Whittaker integrals and underpin the continuity of normalized Fourier and Wave packet transforms on Harish-Chandra type Schwartz spaces. A key methodological theme is reducing to the maximal-parabolic (adjacent/opposite) case using a root-based α-reduction and then transferring MS properties from the basic setting to the general one. The analysis culminates in Harish-Chandra–style radial Casimir calculations and a boundary-term Gauss-divergence argument that control asymptotics and establish the functional equations and normalization required for the Fourier/Wave-packet theory.

Abstract

We give a complete proof of the Maass-Selberg relations for Whittaker integrals on a real reductive group. These relations were asserted In unpublished work of Harish-Chandra, and proven in the basic setting of maximal parabolic parabolic subgroups. Our proof is a reduction to the mentioned basic setting. It makes use of the action of standard intertwining operators on Whittaker distribution vectors in the generalized principal series of representations. The Maass-Selberg relations imply regularity of normalized Whittaker integrals. This is crucial for decent behavior of Fourier and wave packet transforms on the level of spherical Schwartz spaces.

Paper Structure

This paper contains 18 sections, 104 theorems, 384 equations.

Table of Contents

  1. Introduction
  2. Definition of the map $j(Q, \sigma, \nu)$
  3. Interaction with the Weyl group
  4. The B-matrices, reduction arguments
  5. Reduction to maximal parabolic subgroups
  6. Smoothness of $J$
  7. A useful integral formula
  8. Comparison of $J$ with $J^{(\alpha)}$
  9. Comparison of $B$ with $B^{(\alpha)}$, proof of Lemma \ref{['l: comparison B and B ga']}
  10. The C-functions and the Maass--Selberg relations
  11. Maass--Selberg relations in the basic setting
  12. The radial part of the Casimir operator
  13. A result of Harish-Chandra
  14. The normalized Whittaker integral
  15. Fourier transform and Wave packets
  16. ...and 3 more sections

Key Result

Lemma 1.1

$G$ is a finite union of double cosets of the form $N_0 v Q,$ for $v \in N_K({\mathfrak a}).$ The coset $N_0 v Q$ is open in $G$ if and only if $vN_Qv^{-1} \subset \bar{N}_0,$ which in turn is equivalent to the condition that $v Qv^{-1}\in \bar{{\mathcal{P}}}_{\rm st}$.

Theorems & Definitions (122)

  • Lemma 1.1
  • Lemma 1.2
  • Corollary 1.3
  • Definition 1.4
  • Lemma 1.5
  • Corollary 1.6
  • Lemma 1.7
  • Corollary 1.8
  • Lemma 1.9
  • Definition 1.10
  • ...and 112 more