# Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable \$A_r\$ | Annals of Mathematics

Abstract 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⩽5 is also established.

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### 数学年刊（Annals of Mathematics）

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