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Department of Mathematics,
University of California San Diego


Number Theory Seminar - Math 209

Nolan Wallach

UC San Diego

The Whittaker Inversion Theorem and some applications



The Whittaker Plancherel theorem appeared as Chapter 15 in my two volume book, Real Reductive Groups. It was meant to be an application of Harish-Chandra’s Plancherel Theorem.  As it turns out, there are serious gaps in the proof given in the books. At the same time as I was doing my research on the subject, Harish-Chandra was also working on it. His approach was very different from mine and appears as part of Volume 5 of his collected works; which consists of three pieces of research by Harish-Chandra that were incomplete at his death and organized and edited by Gangolli and Varadarajan. Unfortunately,  it also does not contain a proof of the theorem. There was a complication in the proof of this result that caused substantial difficulties which had to do with the image of the analog of Harish-Chandra’s method of descent. In this lecture I will explain how one can complete the proof using a recent result of Raphael Beuzzart-Plessis. I will also give an application of the result to the Fourier transforms of automorphic functions at cusps.

(This seminar will be given remotely, but there will still be a live audience in the lecture room.)

[pre-talk at 1:20PM]

Host: Kiran S. Kedlaya

January 26, 2023

2:00 PM

APM 6402 and Zoom