TY - JOUR

T1 - Weyl group multiple Dirichlet series III

T2 - Eisenstein series and twisted unstable Ar

AU - Brubaker, B.

AU - Bump, D.

AU - Friedberg, S.

AU - Hoffstein, J.

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2007/7

Y1 - 2007/7

N2 - 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.

AB - 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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U2 - 10.4007/annals.2007.166.293

DO - 10.4007/annals.2007.166.293

M3 - Article

AN - SCOPUS:38349056778

VL - 166

SP - 293

EP - 316

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -