论文标题
Dunkl内核和二面体组的交织操作员
The Dunkl kernel and intertwining operator for dihedral groups
论文作者
论文摘要
与有限反射组相关的DUNKL操作员产生了差分差异操作员的交换代数。存在一个唯一的线性运算符，称为交织运算符，该操作员在该代数和标准差分运算符的代数之间交织在一起。在这种上下文中，也存在对傅立叶变换的概括，称为dunkl变换。在本文中，我们确定了dunkl内核的积分表达，这是dunkl变换的整体内核。我们还基于对所有反射组有效的观测值确定在二面基组的情况下，在二面基组的情况下确定一个相互构图的表达式。作为一种特殊情况，我们恢复了[XU，与二面群相关的交织运算符的结果。约束。大约2019]。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使用整体内核和单纯形之间的链接。
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, Intertwining operators associated to dihedral groups. Constr. Approx. 2019]. Crucial in our approach is a systematic use of the link between both integral kernels and the simplex in a suitable high dimensional space.
