Geometric invariants and model theory for singular unitary representations

在本研究项目中，我们研究了用几何方法实现李群的奇异不可约酉表示。首先，通过将基本表示的张量积分解为所讨论表示的几何不变量，我们建立了Dvorsky-Sahi理论的Fock模型版本，它是关于约化李群的奇异么正最高权表示的theta对偶对应的推广。其次，利用根系统上的数据刻画了实4阶例外单李群的四元数结构所产生的准齐次向量空间中的奇异轨道，并证明了Gross和Wallach给出的奇异四元表示可以通过相应的四元素幂零K轨道的几何量子化来实现。

In this research project, we investigated realization of singular irreducible unitary representations of Lie groups through geometric approach. First, we have established the Fock model version of Dvorsky-Sahi theory on an extension of the theta duality correspondence for singular unitary highest weight representations of reductive Lie groups, by decomposing tensor products of fundamental representations in terms of geometric invariants for representations in question. Second, the singular orbits in prehomogeneous vector spaces arising from quaternionic structure of exceptional simple Lie groups of real rank 4 have been described by using data on root systems, and we have proved that the singular quarternionic unitary representations, due to Gross and Wallach, can be realized by geometric quantization of the corresponding quarternionic nilpotent K-orbits.