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

TL;DR

This paper presents a new proof for the existence of Whittaker functionals on $ ext{GL}(n, ext{R})$ using distribution theory and integration by parts, extending local distributions globally and applying to Jacquet integrals and Bessel functions.

Contribution

It introduces a novel approach to establish Whittaker functional existence via distribution extension and birational maps, providing an alternative proof of analytic continuation.

Findings

Distribution realization of Whittaker functionals

Extension of distributions from open Schubert cells to entire group

Application to analytic continuation of Jacquet integrals

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 $\text{GL}(n,\mathbb{R})$.