The Fourier transform for triples of quadratic spaces
[La transformation de Fourier pour les triplets d’espaces quadratiques]
Getz, Jayce R.1 ; Hsu, Chun-Hsien2
1Department of Mathematics, Duke University, Durham, NC 27708 (USA)
2Department of Mathematics, University of Chicago, Chicago, IL 60637 (USA)
Annales de l'Institut Fourier, Online first, 74 p.

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.

Soit $V_1, V_2, V_3$ un triplet d’espaces vectoriels de dimension paire sur un corps de nombres $F$ équipés de formes quadratiques non dégénérées $\mathcal{Q}_1, \mathcal{Q}_2, \mathcal{Q}_3$, respectivement. Soit $Y \subset \prod _{i=1}^3 V_i$ le sous-schéma fermé constitué de $(v_1, v_2, v_3)$ tels que $\mathcal{Q}_1(v_1) = \mathcal{Q}_2(v_2) = \mathcal{Q}_3(v_3)$. On dispose d’une formule sommatoire de Poisson pour ce schéma sous des hypothèses appropriées concernant les fonctions impliquées, mais la transformation de Fourier pertinente n’était auparavant définie que comme une correspondance. Dans le présent article, nous employons un argument global-local novateur pour prouver que cette transformation de Fourier est bien définie sur l’espace de Schwartz de $Y(\mathbb{A}_F)$. Pour exécuter cet argument global-local, nous introduisons des termes de frontière et étendons ainsi la formule sommatoire de Poisson à une classe plus large de fonctions de test. C’est la première fois qu’une formule avec des termes de frontière est prouvée pour une variété sphérique qui n’est pas un espace de Braverman–Kazhdan.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3765
Classification : 11F70, 11F27, 11F66
Keywords: Fourier transform, Poisson summation conjecture, spherical varieties.
Mots-clés : Transformation de Fourier, conjecture sommatoire de Poisson, variétés sphériques.

Getz, Jayce R.  1   ; Hsu, Chun-Hsien  2

1 Department of Mathematics, Duke University, Durham, NC 27708 (USA)
2 Department of Mathematics, University of Chicago, Chicago, IL 60637 (USA) 
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Getz, Jayce R.; Hsu, Chun-Hsien. The Fourier transform for triples of quadratic spaces. Annales de l'Institut Fourier, Online first, 74 p.

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