一类新的超复解析尖点形式
A new class of hypercomplex analytic cusp forms
摘要：
本文研究了Ahlfors-Vahlen群的算术子群上的一类新的Clifford代数值自守形式.我们考虑的形式是在一些甚至$k \in {\mathbb{Z}}$的算子$D Δ^{k/2}$的核心。它们将被称为$k$ --全纯克利福德自守形式。$k$ -全纯Cliffordian函数是一个具有许多函数理论工具的函数。此外，实分量函数还具有它们是齐次和非齐次Weinstein方程的解的性质。这个函数类包含了一组$k$ -超正则函数作为一个特殊的子集。虽然我们还没有能够提出一个建设非零$k$ -超单尖点形式的$k \neq 0$，我们能够做到这一点，在这个更大的功能。在解释了它们与双曲调和自守形式的一般关系之后，我们转向庞加莱级数的构造。这些为我们提供了这个函数类中尖点形式的重要示例。然后我们建立了$k$ -全纯Clifford自守形式空间关于$k$ -超正则Eisenstein级数空间与$k$ -全纯Clifford尖点形式空间的直正交和的分解定理.
英文摘要：
In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator $D Δ^{k/2}$ for some even $k \in {\mathbb{Z}}$. They will be called $k$-holomorphic Cliffordian automorphic forms. $k$-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions have also the property that they are solutions to the homogeneous and inhomogeneous Weinstein equation. This function class includes the set of $k$-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing $k$-hypermonogenic cusp forms for $k \neq 0$, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms we turn to the construction of Poincaré series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of $k$-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of $k$-hypermonogenic Eisenstein series and of $k$-holomorphic Cliffordian cusp forms.
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