The spherical Whittaker Plancherel Theorem and the quantum non-periodic Toda Lattice

This paper is partially an exposition of the method of the proof of the continuous part of the general distributional Whittaker Plancherel Theorem in the special case of the spherical spectrum. It is also an explanation of how this result solves the quantum non-periodic Toda lattices. Combining the ideas involved in both of these results the paper also gives a new reduction of the calculation of spherical Whittaker functions to split groups over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℝ</mml:mi> </mml:math> . It concludes with a new proof of an explicit functional equation which is used in the surjectivity result in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> Plancherel Theorem and an explicit isomorphism theorem for the Whittaker Schwartz Space.