Representations of real reductive Lie groups

实数还原李群的表示

• 批准号: 10640153
• 负责人: MATUMOTO Hisayosi
• 金额: $ 1.02万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640153/
• 关键词: real reductive Lie groups; Unitary representations; degenerate series; derived functor modules; 退化系列表現; 既約ユニタリ表現; 分岐則; 半単純リー群; ホイタッカーモデル

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.

本学年，我主要研究了实还原李群的简并主级数，得到了以下结论。我们考虑 SO(m, n)（分别为 U(m, n)）的最大抛物线子群，使其列维部分同构于 SO(m - n) x GL(n,R)（分别为 U(m) - n) x GL(n, C))。我们考虑由来自 SO(m - n) 的不可约有限维表示的抛物线子群的表示导出的 SO(m, n) (分别为 U(m, n)) 的表示（分别为 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)（分别是 U(m, 1)）的限制所获得的表示的可还原性。今年，我得到了不可约的结果。对于 U(m, n) 的情况和 SO(m, n) 的“足够”正的情况”，除上述之外没有其他可约性。对于 SO(m, n) 的情况，情况相当事实上，Farmar 在 SO(3, 2) 的情况下发现了在最奇异参数处的额外可归约性。我们的可归约性是用简并主级数的 K 型分解来描述的，它与限制相兼容。到更小的 SO(m, k) (k < n)，我们可以获得一些作为不可约成分出现的派生函子模块的分支规则。