Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

• 成果类型: Article
• 署名作者: Brubaker, Ben; Bump, Daniel; Friedberg, Solomon
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.2.13
• 发表日期: 2011
• 页码: 1081-1120
• 关键词: whittaker functions; canonical bases; combinatorics; characters; a(r); gln

摘要

We show that the Whittaker coefficients of Borcl Eisenstein series on the metaplectic covers of 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.