Poisson summation conjecture on Braverman-Kazhdan spaces

Abstract

Braverman and Kazhdan proposed a conjecture, later refined by Ng\^{o} and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. In the dissertation, we develop local Fourier theory and give explicit formulae for Fourier transforms on Braverman–Kazhdan spaces attached to maximal parabolic subgroups of split, almost simple, simply connected groups. In the nonarchimedean setting, we also give explicit representation theoretic descriptions of Schwartz spaces and verify several conjectural properties of Schwartz spaces. Part of the thesis is based on joint work with Jayce Getz and Spencer Leslie.