AI 中文总结
本文引入并研究Cartan运动群的代数Dirac算子，建立其基本性质与Dirac上同调，证明Vogan猜想的强形式，明确确定Dirac系列，并解释其作为代数形变族Dirac理论零纤维的获得方式。
AI 中文摘要
我们引入并研究了与实约化李群相关联的Cartan运动群的代数Dirac算子。我们建立了其基本性质，包括一个涉及自然Casimir型元素的平方公式，定义了相应的Dirac上同调概念，并证明了Vogan猜想的一个强形式：Dirac系列中的模由其Dirac上同调完全决定。我们明确确定了Cartan运动群的Dirac系列。我们解释了Cartan运动群的Dirac理论如何作为最近引入的相应实约化群代数形变族的Dirac理论的零纤维而获得。
英文摘要
We introduce and study the algebraic Dirac operator for the Cartan motion group associated with a real reductive Lie group. We establish its fundamental properties, including a square formula involving a natural Casimir-type element, define the corresponding notion of Dirac cohomology, and prove a strong form of Vogan's conjecture: a module in the Dirac series is completely determined by its Dirac cohomology. We explicitly determine the Dirac series of a Cartan motion group. We explain how the Dirac theory of a Cartan motion group can be obtained as the zero fiber of the recently introduced Dirac theory for the algebraic deformation family of the corresponding real reductive group.
