Bhargava Integer Cubes and Weyl Group Multiple Dirichlet Series

We investigate the Shintani zeta functions associated to the prehomogeneous spaces, the example under consideration is the set of 2 × 2 × 2 integer cubes. We show that there are three relative invariants under a certain parabolic group action, they all have arithmetic nature and completely determine the equivalence classes. We show that the associated Shintani zeta function coincides with the A3 Weyl group multiple Dirichlet series. Finally, we show that the set of semi-stable integer orbits maps finitely and surjectively to a certain moduli space.