Weyl group multiple Dirichlet series, Eisenstein series and crystal bases
Weyl group multiple Dirichlet series, Eisenstein series and crystal bases
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Weyl群多重狄利克雷级数、爱森斯坦级数和晶体基
DOI：
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发表时间：
2011
期刊：
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通讯作者：
S. Friedberg
中科院分区：
文献类型：
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作者：
Ben Brubaker;D. Bump;S. Friedberg
我们证明了GLr+1的亚代数覆盖上Borel Eisenstein级数的Whittaker系数可以描述为r个复变量的重Dirichlet级数，其系数通过在晶体图的每个顶点上附加一个数论量（高斯和的乘积）来计算.这些高斯和依赖于字符串数据”之前介绍的工作Lusztig，Berenstein和Zelevinsky，和Littelmann。这些数据是从给定顶点到最低权重顶点的路径中的段的长度，取决于长Weyl群元素到简单反射的因式分解。系数也可以描述为严格Gelfand-Tsetlin模式上的和。该描述是统一的次复盖的程度。
We show that the Whittaker coecients of Borel Eisenstein series on the metaplectic covers of GLr+1 can be described as multiple Dirichlet series in r complex variables, whose coecients 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 reections. The coecients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.