Representations of real reductive Lie groups

本学年主要研究了真实的约化李群的退化主列，得到了以下结果。本文考虑SO（m，n）的极大抛物子群（分别为. U（m，n））使得它的Levi部分同构于SO（m-n）xGL（n，R）（resp. U（m-n）xGL（n，C））。本文考虑由SO（m-n）（resp.）的不可约有限维表示的抛物子群表示导出的SO（m，n）（resp.）U（m-n）和GL（n，R）的一维表示（resp. GL（n，R））。在上一学年，我发现了一个约化的表示得到的限制SO（m，l）（分别。U（m，1））。这一年，我得到了一个不可还原的结果。对于U（m，n）的情形和SO（m，n）的“充分”正情形”，除上述情形外，没有任何归约。对于SO（m，n）的情况，情况是相当微妙的。事实上，Farmar已经找到了SO（3，2）在最奇异参数处的一个额外的约化，我们的约化是用退化主级数的K-型分解来描述的。它与对较小SO（m，k）（k < n）的限制相容，并且我们可以得到某些作为不可约分支出现的导函子模的分支规则。

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.