补级数的结构与特殊表示
The structure of complementary series and special representations
摘要：
本文综述了SU（n，1）和SO（n，1）群的几种不可约余级数表示模型及其极限、特殊表示，包括新的表示。这些群的几何意义是众所周知的，它们穷尽了单位表示在不可约酉表示空间中不是孤立的单李群的列表（即，不具有Kazhdan性质），因此存在这些群的不可约酉表示--所谓的"特殊表示“--对于这些表示中系数的群的第一上同调是非平凡的。由于技术原因，考虑群O（n，1）和U（n，1）更方便。本文的大部分内容都是关于群U（n，1）的。主要的重点是所谓的交换模型的特殊和互补的系列表示：在这些模型中，最大幂幺子群表示的乘法器的情况下，O（n，1），和规范模型的海森堡表示的情况下，U（n，1）。之前，仅对$SL(2,\RR)$组的这些模型进行了研究。它们对于实现当前群的非局部表示特别重要，这将在其他地方讨论。我们充分利用SO（n，1）不可约表示的“密度”：它们对极大抛物子群$ {P} $的限制是等价的不可约表示.相反，为了将${P}$的不可约表示扩展为SO（n，1）的表示，我们必须另外定义仅一个对合。对于群U（n，1），情况类似，但稍微复杂一些。
英文摘要：
We give a survey of several models of irreducible complementary series representations and their limits, special representations, for the groups SU(n,1) and SO(n,1), including new ones. These groups, whose geometrical meaning is well known, exhaust the list of simple Lie groups for which the identity representation is not isolated in the space of irreducible unitary representations (i.e., which do not have the Kazhdan property) and hence there exist irreducible unitary representations of these groups -- so-called ``special representations'' -- for which the first cohomology of the group with coefficients in these representations is nontrivial. By technical reasons, it is more convenient to consider the groups O(n,1) and U(n,1). Most part of the paper is devoted to the group U(n,1). The main emphasis is on the so-called commutative models of special and complementary series representations: in these models, the maximal unipotent subgroup is represented by multiplicators in the case of O(n,1), and by the canonical model of the Heisenberg representations in the case of U(n,1). Earlier, these models were studied only for the group $SL(2,\RR)$. They are especially important for realization of nonlocal representations of current groups, which will be considered elsewhere. We substantially use the ``density'' of the irreducible representations under study of SO(n,1): their restrictions to the maximal parabolic subgroup $ {P} $ are equivalent irreducible representations. Conversely, in order to extend an irreducible representation of ${P}$ to a representation of SO(n,1), we must additionally define only one involution. For the group U(n,1), the situation is similar but slightly more complicated.
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