Weyl Group Multiple Dirichlet Series III: Eisenstein Series and Twisted Unstable Ar

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums. The case where n is small is harder, and is addressed in this paper when Φ = Ar. “Twisted” Dirichet series are considered, which contain the series of [4] as a special case. These series are not Euler products, but due to the twisted multiplicativity of their coefficients, they are determined by their p-parts. The p-part is given as a sum of products of Gauss sums, parametrized by strict Gelfand-Tsetlin patterns. It is conjectured that these multiple Dirichlet series are Whittaker coefficients of Eisenstein series on the n-fold metaplectic cover of GLr+1, and this is proved if r = 2 or n = 1. The equivalence of our definition with that of Chinta [11] when n = 2 and r 6 5 is also established. AMS Subject Classification (2000):11F30 (Primary), 11F27, 20F55 (Secondary). Let F be a totally complex algebraic number field containing the group μ2n of 2n-th roots of unity. Thus −1 is an n-th power in F . Let Φ ⊂ R be a reduced root system. It has been shown in Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] how one can associate a multiple Dirichlet series with Φ; its coefficients involve n-th order Gauss sums. A condition of stability is imposed in this definition, which amounts to n being sufficiently large, depending on Φ. In this paper we will propose a description of the Weyl group multiple Dirichlet series in the unstable case when Φ has Cartan type Ar, and present the evidence in support of this description. We will refer to this as the Gelfand-Tsetlin description. A striking characteristic of this description is that it gives a single formula valid for all n for these coefficients, that reduces to the stable description when n is sufficiently large. We conjecture that this Weyl group multiple Dirichlet series coincides with the Whittaker coefficient of an Eisenstein series. The Eisenstein series E(g; s1, · · · , sr) is of minimal parabolic type, on an n-fold metaplectic cover of an algebraic group defined over F whose root system is the dual 1Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA. email: [email protected] 2Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA. email: [email protected] 3Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA. email: [email protected] 4Department of Mathematics, Brown University, Providence, RI 02912, USA. email: [email protected]