Department of Mathematics, University of California at San Diego, La Jolla, CA 92093.
Proc Natl Acad Sci U S A. 1983 Nov;80(22):7047-50. doi: 10.1073/pnas.80.22.7047.
For a real semisimple Lie group G, the description of the unitary dual remains an elusive question. One of the difficulties has been the lack of technique for constructing unitary representations. Unitary induction from parabolic subgroups of G yields unitary representations by the very definition of these representations. However, not all unitary irreducible representations of G are obtained by this type of induction. In addition, we need derived functor parabolic induction [ef. Vogan, D. (1981) Representations of Real Reductive Lie Groups (Birkhäuser, Boston)] to describe all irreducible representations of G. For this second type of induction, the obvious analogues from parabolic subgroup induction regarding unitarity are false. In this announcement, we describe a setting where derived functor parabolic induction yields unitary representations of G. These results include proofs of unitarity for some of the representations conjectured to be unitary by Vogan and Zuckerman [(1983) Invent. Math., in press] and also proofs of unitarity for some which lie outside the domain described in those conjectures.
对于一个真正的半单李群 G，其酉对偶的描述仍然是一个难以捉摸的问题。其中一个困难是缺乏构造酉表示的技术。通过这些表示的定义，从 G 的抛物子群进行酉诱导可以得到酉表示。然而，并非 G 的所有酉不可约表示都可以通过这种类型的诱导得到。此外，我们需要导出函子抛物诱导[ef。Vogan，D.（1981）实约化李群的表示（Birkhäuser，波士顿）]来描述 G 的所有不可约表示。对于第二种类型的诱导，从抛物子群诱导关于酉的明显类似物是错误的。在这个公告中，我们描述了一个导出函子抛物诱导产生 G 的酉表示的环境。这些结果包括对 Vogan 和 Zuckerman[（1983）Invent。Math.，in press]猜测的一些酉表示的酉性的证明，以及对那些在这些猜测描述之外的表示的酉性的证明。
