A summation formula for triples of quadratic spaces II
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 \begin{align*} Y \subset \prod_{i=1}V_i \end{align*} be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. The first author and B. Liu previously proved a Poisson summation formula for this scheme under suitable assumptions on the functions involved. In the current work we extend the formula to a broader class of test functions which necessitates the introduction of boundary terms. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a BravermanKazhdan space. As an application, we prove that the Fourier transform on $Y,$ previously defined as a correspondence, descends to an isomorphism of the Schwartz space of $Y.$
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.11490
 Bibcode:
 2020arXiv200911490G
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory;
 11F70 (Primary) 11F66 (Secondary)
 EPrint:
 76 pages. Changed notation $X_P$ to $X^\circ$. Strengthened Lemma 5.7, Proposition 11.3, and Theorem 12.1. Modified proofs of Propositions 3.12, 3.16