Whittaker Coefficients of Metaplectic Eisenstein Series
Whittaker Coefficients of Metaplectic Eisenstein Series
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Metaplectic Eisenstein 级数的 Whittaker 系数
DOI：
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发表时间：
2014
期刊：
影响因子：
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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;S. Friedberg
研究了分裂约化群的广义复盖上极大抛物型爱森斯坦级数的Whittaker系数。根据爱森斯坦级数理论，这些系数具有亚纯延拓和泛函方程。然而，它们不是欧拉的，在简化情况下计算它们的标准方法不适用于覆盖。对于“共微”极大抛物，我们给出了系数的显式描述为算术内容用指数和表示的狄利克雷级数。然后证明指数和满足扭曲乘法性，将其确定为素数幂贡献。反过来，利用Kamnitzer的结果，这些数据与基于对偶群的正则基的Lusztig数据联系起来。对于一般的Lusztig数据，幂指数和被证明是简化的。在其余的退化情况下，指数和似乎最好用高斯和来表示，这取决于正则基的字符串数据，如GL4中的详细示例所示。因此，我们证明了对于正则基，广义Whittaker系数的算术部分与这两个表达式之间的关系密切相关。
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 GL4. Thus we demonstrate that the arithmetic part of metaplectic Whittaker coefficients is intimately connected to the relations between these two expressions for canonical bases.