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

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 These relations were asserted In unpublished work of Harish-Chandra, and proven in the basic setting of maximal parabolic parabolic subgroups.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