Geodesic
Weyl group multiple Dirichlet series, Eisenstein series and crystal bases
Ben Brubaker1 ; Daniel Bump2 ; Solomon Friedberg3
1 Massachusetts Institute of Technology<br/> Cambridge, MA
2 Stanford University<br/> Stanford, CA
3 Boston College<br/> Chestnut Hill, MA
Annals of mathematics, Tome 173 (2011) no. 2, pp. 1081-1120.

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We show that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of ${\rm GL}_{r+1}$ 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.
DOI : 10.4007/annals.2011.173.2.13

Ben Brubaker 1 ; Daniel Bump 2 ; Solomon Friedberg 3

1 Massachusetts Institute of Technology<br/> Cambridge, MA
2 Stanford University<br/> Stanford, CA
3 Boston College<br/> Chestnut Hill, MA 
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Ben Brubaker; Daniel Bump; Solomon Friedberg. Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. Annals of mathematics, Tome 173 (2011) no. 2, pp. 1081-1120. doi : 10.4007/annals.2011.173.2.13. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.2.13/

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