Whittaker Coefficients of Metaplectic Eisenstein Series

被引:9
作者
Brubaker, Benjamin [1 ]
Friedberg, Solomon [2 ]
机构
[1] Univ Minnesota, Sch Math, Minneapolis, MN 55455 USA
[2] Boston Coll, Dept Math, Chestnut Hill, MA 02467 USA
基金
美国国家科学基金会;
关键词
Eisenstein series; Metaplectic group; Whittaker coefficient; Canonical bases; Lusztig data; MULTIPLE DIRICHLET SERIES;
D O I
10.1007/s00039-015-0329-4
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However they are not Eulerian and the standard methods to compute them in the reductive case do not apply to covers. For "cominuscule" maximal parabolics, we give an explicit description of the coefficients as Dirichlet series whose arithmetic content is expressed in an exponential sum. The exponential sum is then shown to satisfy a twisted multiplicativity, reducing its determination to prime power contributions. These, in turn, are connected to Lusztig data for canonical bases on the dual group using a result of Kamnitzer. The exponential sum at prime powers is shown to simplify for generic Lusztig data. At the remaining degenerate cases, the exponential sum seems best expressed in terms of Gauss sums depending on string data for canonical bases, as shown in a detailed example in GL (4). Thus we demonstrate that the arithmetic part of metaplectic Whittaker coefficients is intimately connected to the relations between these two expressions for canonical bases.
引用
收藏
页码:1180 / 1239
页数:60
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