TY - JOUR

T1 - Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

AU - Brubaker, Ben

AU - Bump, Daniel

AU - Friedberg, Solomon

PY - 2011/3

Y1 - 2011/3

N2 - We show that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of GLr+1 can be described as multiple Dirichlet series in r complex variables, whose coefficients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on "string data" previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reflections. The coefficients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

AB - We show that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of GLr+1 can be described as multiple Dirichlet series in r complex variables, whose coefficients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on "string data" previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reflections. The coefficients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

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U2 - 10.4007/annals.2011.173.2.13

DO - 10.4007/annals.2011.173.2.13

M3 - Article

AN - SCOPUS:79953206671

SN - 0003-486X

VL - 173

SP - 1081

EP - 1120

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -