Schubert cells and Whittaker functionals for GL(r, ℝ), part II: Existence via integration by parts

We give a new proof of the existence of Whittaker functionals for principal series representation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> GL(r,R) , utilizing the analytic theory of distributions. We realize Whittaker functionals as equivariant distributions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> GL(r,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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> GL(r,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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> GL(r,R) .