Representations of real reductive Lie groups

在这一学年里，我主要研究了实约化李群的退化主级数列，得到了以下结果。本文考虑SO(m，n)的一个极大抛物子群。U(m，n))，使得它的Levi部分同构于SO(m-n)x GL(n，R)(分别.U(m-n)x GL(n，C))。本文从SO(m-n)的不可约有限维表示出发，考虑了由抛物子群的表示导出的SO(m，n)(分别为U(m，n))的表示。U(m-n)和GL(n，R)的一维表示(分别为Gl(n，R))。在上一学年，我发现通过考虑对SO(m，L)的限制而得到的表示是可约化的。U(m，1))。在这一年里，我取得了一个不可逆转的成绩。对于U(m，n)的情形和SO(m，n)的“充分”正情形，除上述情形外，不存在可约性。对于SO(m，n)的情况，情况是相当微妙的。事实上，Farmar已经在SO(3，2)的最奇异参数处发现了一个额外的可约性。我们的可约性用退化主级数的K型分解来描述。它与较小的SO(m，k)(k&lt；n)的限制是相容的，并且可以得到一些作为不可约分量出现的派生函子模的分枝规则。

In this academic year, I have been studied mainly on degenerate principal series of real reductive Lie groups and obtained the following. We consider a maximal parabolic subgroup of SO(m, n) (resp. U(m, n)) such that its Levi part is isomorphic to SO(m - n) x GL(n,R) (resp. U(m - n) x GL(n, C)). We consider the representations of SO(m, n) (resp, U(m, n)) induced from the representations of the parabolic subgroup coming from irreducible finite-dimensional representations of SO(m - n) (resp. U(m - n))) and one-dimensional representation of GL(n, R) (resp. GL(n, R)). In the last academic year, I found a reducibility of the representation obtained by considering the restriction to SO(m, l) (resp. U(m, 1)). In this year, I obtained an irreducibilty result. For the case of U(m, n) and the "sufficitintly" positive case" of SO(m, n), there is no reducibility other than the above. For the case of SO(m, n), the situation is quite subtle. In fact, Farmar had found an extra reducibility at the most singular parameter for the case of SO(3, 2).Our reducibility is described in terms of K-type decomposition of the degenerate principal series. It is compatible with the restriction tosmaller SO(m, k) (k < n) and we can obtain branching rule of some derived functor modules which appear as irreducible constituents.