Schubert cells and Whittaker functionals for $\text{GL}(n,\mathbb{R})$ part II: Existence via integration by parts

Abstract

We give a new proof of the existence of Whittaker functionals for principal series representation of $\text{GL}(n,\mathbb{R})$, utilizing the analytic theory of distributions. We realize Whittaker functionals as equivariant distributions on $\text{GL}(n,\mathbb{R})$, whose restriction to the open Schubert cell is unique up to a constant. Using a birational map on the Schubert cells, we show that the unique distribution on the open Schubert cell extends to a distribution on the entire space $\text{GL}(n,\mathbb{R})$. This technique gives a proof of the analytic continuation of Jacquet integrals via integration by parts. We briefly discuss an application of the method to the Bessel functions on $\text{GL}(n,\mathbb{R})$.