Farid Hosseinijafari of Duke : $\rho$-Schwartz Spaces and Modulation Groups

Posted by Ramin Takloo-Bighash , part of the Number Theory Seminar.

At

Oct. 12, 2026, 9 a.m.

In

512 SEO

Abstract

Braverman–Kazhdan and Ngô conjectured the existence of a $\rho$-Schwartz space attached to a representation of the $L$-group of a reductive group over a local field $F$, together with a suitable notion of Fourier transform. We construct the $\rho$-Fourier transform over arbitrary local fields. Over non-Archimedean fields, we construct the corresponding Schwartz space, while in the Archimedean case we construct an approximation to it. We then explain how Schwartz spaces associated with certain spherical varieties naturally carry representations of groups that we call modulation groups. This provides a broad generalization of the defining representation of the metaplectic group. In this talk, I will go over both parts of the project, beginning with some background from classical Fourier analysis. This is joint work with J. Getz, A. Gutiérrez Terradillos, S. Lee, A. Slipper, M.-H. Tomé, G. Xi, H. Yao, and A. Zhao.