# 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.

## Authors

Ben Brubaker

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States

Daniel Bump

Department of Mathematics, Stanford University, Stanford, CA 94305, United States

Solomon Friedberg

Department of Mathematics, Boston College, Chestnut Hill, MA 02467, United States

Jeffrey Hoffstein

Department of Mathematics, Brown University, Providence, RI 02912, United States