Weyl group multiple Dirichlet series III:: Eisenstein series and twisted unstable Ar

Weyl group multiple Dirichlet series were associated with a root system Phi 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 Phi = A(r). "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 GL(r+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 <= 5 is also established.