##### Department of Mathematics,

University of California San Diego

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### Number Theory Seminar - Math 209

## Nolan Wallach

#### UC San Diego

## The Whittaker Inversion Theorem and some applications

##### Abstract:

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

See https://www.math.ucsd.edu/~nts

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