Multiple Dirichlet series associated with quadrics

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.