Multiple Dirichlet series associated with quadrics

TL;DR

The paper develops a three-variable zeta framework for the zero loci of quadratic forms, i.e., quadrics, by embedding the problem into a filtration of a quadratic space and employing a refined Poisson summation on singular varieties. It combines the Weil representation, integral transforms, and orbital-integral analysis to relate zeta integrals $Z(f,s_1,s_2,w)$ to the three-parameter Dirichlet series $\xi_{\pm}(s_1,s_2,w)$, establishing analytic continuation and a fundamental functional equation $Z(f,s_1,s_2,w)=Z(\hat f,s_1,s_2,2-s_1-s_2-w)$. The main contribution is proving the third type of functional equation, alongside the known $s_i$-type equations, which together realize the $A_3$ Weyl-group symmetry in this quadrics setting. This extends Shintani's and Braverman–Kazhdan-inspired Poisson-summation techniques to a non-prehomogeneous quadric context, linking orbit counts to Weyl-group multiple Dirichlet series and offering a framework with potential connections to metaplectic Eisenstein series and adelic harmonic analysis on spherical varieties.

Abstract

We define a multiple Dirichlet series associated with quadrics which is the zero locus of a quadratic form. This multiple Dirichlet series is linked to a Shintani zeta function associated with a prehomogeneous vector space. To obtain the functional equations we construct a filtration of the quadratic space and define the parabolic group actions, and then apply a non-abelian Poisson summation formula which sums over all lower dimensional quadrics along with the original quadrics. We show the group of functional equations is isomorphic to a finite Weyl group of type A3.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.6
• Proposition 2.7
• Lemma 2.8