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 -