Journal of Lie Theory

no. 1
The spherical Whittaker Plancherel Theorem and the quantum non-periodic Toda Lattice
Wallach, Nolan R.1
1Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 307-338

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 $\mathbb{R} $. It concludes with a new proof of an explicit functional equation which is used in the surjectivity result in the $L^{2}$ Plancherel Theorem and an explicit isomorphism theorem for the Whittaker Schwartz Space.

Received:
Accepted:
Published online:
DOI: 10.5802/jolt.1427
Classification: 20E45, 22E30, 22E70, 43A85
Keywords: Whittaker vector, Plancherel theorem, Inversion theorem

Wallach, Nolan R.  1

1 Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA 
Wallach, Nolan R. The spherical Whittaker Plancherel Theorem and the quantum non-periodic Toda Lattice. Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 307-338. doi: 10.5802/jolt.1427
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