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