Weyl Group Multiple Dirichlet Series: Some Open Problems

The problems described here concern Weyl group multiple Dirichlet series (WMDs). Let F be a totally complex algebraic number field containing the group μn of n-th roots of unity. We will assume that −1 is an n-th power in F . Let Φ ⊂ R be a reduced root system. Then one can attach to Φ and n a family of multiple Dirichlet series whose coefficients involve products of n-th order Gauss sums. These series are the Whittaker coefficients of the metaplectic Eisenstein series obtained by inducing from the Borel subgroup. (See [10, 11, 13] for aspects of the metaplectic group.) As such, they should have meromorphic continuation in all variables and satisfy a finite group of functional equations. It turns out to be difficult to study these Whittaker coefficients directly. However, we believe that one can write down the correct objects and prove their functional equations by means of a reduction to S̃L2, using Bochner’s theorem on continuation to a tube domain (see [2] or [12] for the S̃L2 theory, and [1] or [9] for Bochner’s theorem). The study of such “Weyl group multiple Dirichlet series” using this method was initiated by Brubaker, Bump, Chinta, Friedberg and Hoffstein, and has been described in a series of papers ([3], [4], [5], [6]). If n is sufficiently large (the ‘stable case’) these series are completely understood, i.e. the meromorphic continuation and functional equations are proved. However, if n is not large (the ‘non-stable case’), the situation is much more complicated. The problem is to understand it fully. Let us next explain what is known in the non-stable case. The multiple Dirichlet series of concern here are of the form