A summation formula 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 ${\mathcal{Q}}_1,{\mathcal{Q}}_2,{\mathcal{Q}}_3$, respectively. Let \begin{align*} Y \subset \prod_{i=1}V_i \end{align*} be the closed subscheme consisting of $(v_1,v_2,v_3)$ on which ${\mathcal{Q}}_1(v_1)={\mathcal{Q}}_2(v_2)={\mathcal{Q}}_3(v_3)$. Motivated by conjectures of Braverman and Kazhdan and related work of Lafforgue, Ng^o, and Sakellaridis we prove an analogue of the Poisson summation formula for certain functions on this space.