Browsing by Subject "Poisson summation conjecture"

Conjectures of Braverman-Kazhdan, L. Lafforgue, Ngô and Sakellaridis imply that all affine spherical varieties admit generalized Poisson summation formula. In this dissertation we establish a generalized Poisson summation formula for certain spaces of test functions on the zero locus of a quadratic form. The functions are built from the Whittaker coefficients of automorphic representations on GL_n. We also give an expression of the local factors where all the data is unramified.

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.