$k$ -Dirac络合物
$k$-Dirac Complexes
摘要：
这是一系列两篇论文中的第一篇。本文构造了一类特殊类型的$|2|$ -分次抛物几何的齐性空间上的不变微分算子复形。我们称之为$k$ -狄拉克复合物。更明确地说，我们将表明，每个$k$ -狄拉克复杂的出现作为一个相关的BGG序列的直接图像，所以这适合的计划彭罗斯变换。我们还将证明每个$k$ -Dirac复形形式上是精确的，即，它在任何不动点诱导出无穷（加权）喷流的长精确序列。在系列的第二部分中，我们使用这些信息来表明，每个$k$ -狄拉克复杂的是准确的在任何一点上的正式幂级数的水平，它下降到一个决议的$k$ -狄拉克运营商研究克利福德分析。
英文摘要：
This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as the direct image of a relative BGG sequence and so this fits into the scheme of the Penrose transform. We will also prove that each $k$-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series we use this information to show that each $k$-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the $k$-Dirac operator studied in Clifford analysis.
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