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The Dunkl kernel and intertwining operator for dihedral groups
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-01-19 , DOI: 10.1016/j.jfa.2021.108932
Hendrik De Bie , Pan Lian

Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the algebra of standard differential operators. There also exists a generalization of the Fourier transform in this context called Dunkl transform.

In this paper, we determine an integral expression for the Dunkl kernel, which is the integral kernel of the Dunkl transform, for all dihedral groups. We also determine an integral expression for the intertwining operator in the case of dihedral groups, based on observations valid for all reflection groups. As a special case, we recover the result of Xu (2019) [36]. Crucial in our approach is a systematic use of the link between both integral kernels and the simplex in a suitable high dimensional space.



中文翻译:

Dunkl内核和二面体组的交织算子

与有限反射组关联的Dunkl算子生成一个微分-差分算子的交换代数。存在一个称为缠绕算子的独特线性算子,它在该代数和标准微分算子的代数之间交织。在这种情况下,也存在称为Dunkl变换的傅立叶变换的一般化。

在本文中,我们为所有二面体组确定Dunkl核的积分表达式,该表达式是Dunkl变换的积分核。我们还基于对所有反射组均有效的观察结果,确定了二面体组中交织算子的积分表达式。作为特例,我们恢复了Xu(2019)[36]的结果。在我们的方法中,至关重要的是在合适的高维空间中系统地利用积分内核和单纯形之间的链接。

更新日期:2021-01-22
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