The Fourier transform for triples of quadratic spaces

Abstract

Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms ${\mathcal{Q}}_1,{\mathcal{Q}}_2,{\mathcal{Q}}_3$, respectively. Let $Y\subset {\prod }_{i=1}^3{V}_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that ${\mathcal{Q}}_1(v_1)={\mathcal{Q}}_2(v_2)={\mathcal{Q}}_3(v_3)$. One has a Poisson summation formula for this scheme under suitable assumptions on the functions involved, but the relevant Fourier transform was previously only defined as a correspondence. In the current paper we employ a novel global-to-local argument to prove that this Fourier transform is well-defined on the Schwartz space of $Y({\mathbb{A}}_F).$ To execute the global-to-local argument, we introduce boundary terms and thereby extend the Poisson summation formula to a broader class of test functions. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space.

60 pages. Accepted by Annales de l'Institut Fourier