Unipotent Degrees of Imprimitive Complex Reflection Groups
Unipotent Degrees of Imprimitive Complex Reflection Groups
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DOI：
10.1006/jabr.1995.1329
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发表时间：
1995-11
影响因子：
0.9
通讯作者：
G. Malle
中科院分区：
文献类型：
--
作者：
G. Malle
在李型G(Q)有限群的表示理论中，幂等特征标起着基础性的作用。它们的次数被视为Q中的多项式，仅依赖于G(Q)的Weyl群。G.Lusztig(Astérisque 212(1993)191-203)已经证明了人们可以定义一般有限Coxeter群的幂等度。本文对由n个反射生成的n维复反射群的两个无穷级数，构造了一个具有与有限Weyl群的特征标度相同的组合性质的幂等度集。特别地，它们通过傅里叶变换矩阵与伪度相关，并且与Frobenius的适当特征值一起提供了SL2(Z)的表示。
In the representation theory of finite groups of Lie type G (q) the unipotent characters play a fundamental role. Their degrees, seen as polynomials in q, are only dependent on the Weyl group of G (q). G. Lusztig (Astérisque 212 (1993) 191–203) has shown that one can define unipotent degrees for a general finite Coxeter group. In this article we construct, for the two infinite series of n-dimensional complex reflection groups that are generated by n reflections, a set of unipotent degrees, with the same combinatorial properties as the unipotent character degrees of a finite Weyl group. In particular they are related by a Fourier transform matrix to the fake degrees, and together with the appropriate eigenvalues of Frobenius they provide a representation of SL2 (Z).