Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable $A_r$

Abstract

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 $\mathrm{GL}_{r + 1}$, and this is proved if $r = 2$ or $n = 1$. The equivalence of our definition with that of Chinta [11] when $n = 2$ and $r \leqslant 5$ is also established.