Uniform temperedness for Whittaker integrals on 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.