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

TL;DR

This paper provides a complete proof of the Maass-Selberg relations for Whittaker integrals on real reductive groups, crucial for understanding Fourier transforms in representation theory.

Contribution

It offers a new proof of the Maass-Selberg relations by reducing to a basic setting, clarifying their role in the theory of Whittaker functions.

Findings

Proves Maass-Selberg relations for Whittaker integrals on real reductive groups.

Shows regularity of normalized Whittaker integrals.

Facilitates better understanding of Fourier and wave packet transforms.

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 These relations were asserted In unpublished work of Harish-Chandra, and proven in the basic setting of maximal parabolic parabolic subgroups.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.