Schubert cells and Whittaker functionals for GL(r, 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

, utilizing the analytic theory of distributions. We realize Whittaker functionals as equivariant distributions on

GL ⁡ ( r , R ) \operatorname{GL}(r,\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

GL ⁡ ( r , R ) \operatorname{GL}(r,\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

GL ⁡ ( r , R ) \operatorname{GL}(r,\mathbb{R})

.