Representations of real reductive Lie groups

在这一学年中，我主要学习了实数约化李群的退化主级数，得到了以下结论：考虑SO（m, n）的极大抛物子群。U(m, n))使得它的Levi部分同构于SO(m - n) x GL(n，R) (p。U（m - n) x GL(n， C)）本文考虑了由抛物子群的不可约有限维表示引申出的SO(m, n) （resp， U(m, n)）的表示。U(m - n))和GL（n， R）的一维表示（p < 0.05）。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具有可约性。U (m, 1))。在这一年里，我得到了一个不可约的结果。对于U（m, n）和SO（m, n）的“充分”正情况，除上述情况外，没有其他可约性。对于SO（m, n）的情况，情况很微妙。事实上，Farmar在SO（3,2）的情况下，在最奇异参数处发现了额外的可约性。我们的可约性用退化主级数的k型分解来描述。它与小于SO(m, k) （k < n）的限制是相容的，并且我们可以得到一些派生的函子模的分支规则，这些函子模表现为不可约的成分。

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.