On character sheaves and characters of reductive groups at unipotent classes
On character sheaves and characters of reductive groups at unipotent classes
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关于单能类上的特征轮和还原群的特征
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发表时间：
2013
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通讯作者：
J. Michel
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文献类型：
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作者：
Franccois Digne;G. Lehrer;J. Michel
为了确定有限约化群在幂幺元处的特征值，我们证明了一些关于广义Gelfand-Graev特征与特征层的特征函数（这里称为Lusztig函数）的内积的结果。这些是用来确定广义Gelfand-Graev字符的投影到unipotent字符的空间，并与一个给定的波前集字符的空间。这种预测主要是用外尔群数据来表示的。我们展示了如何在他们的单幂支持或波前集字符的值是由这样的数据。在一些特殊的群体，我们表明，一个广义的Gelfand-Graev字符的投影到一个家庭具有相同的波前集是（签署）的对偶梅林变换。利用这些结果，在某些情况下，我们能够确定的根的单位关系几乎字符的特征功能。特别是，我们展示了如何计算所有的幂幺字符的值在所有的幂幺类的特殊伴随群的类型G2，F4，E6，E7和E8。我们还提供了一个附录，给出了所有拟单群上尖点特征层的完整列表。
With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front set are determined by such data. In some exceptional groups we show that the projection of a generalised Gelfand-Graev character to a family with the same wave front set is (up to sign) the dual of a Mellin transform. Using these results, in certain cases we are able to determine roots of unity which relate almost characters to the characteristic functions. In particular we show how to compute the values of all unipotent characters at all unipotent classes for the exceptional adjoint groups of type G2, F4, E6, E7 and E8. We also pro- vide an appendix which gives a complete list of the cuspidal character sheaves on all quasi-simple groups.