The Fourier transform for triples of quadratic spaces

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 𝒬 1 , 𝒬 2 , 𝒬 3 , respectively. Let Y ⊂ ∏ i = 1 3 V i be the closed subscheme consisting of ( v 1 , v 2 , v 3 ) such that 𝒬 1 ( v 1 ) = 𝒬 2 ( v 2 ) = 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 ( 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.