1999 Fiscal Year Final Research Report Summary
Representations of real reductive Lie groups
| Project/Area Number |
10640153
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MATUMOTO Hisayosi Grad. Sch. of Math. Sci., The University of Tokyo, Assoc. Prof., 大学院・数理科学研究科, 助教授 (50272597)
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| Co-Investigator(Kenkyū-buntansha) |
ODA Takayuki Grad. Sch. of Math. Sci., The University of Tokyo, Prof., 大学院・数理科学研究科, 教授 (10109415)
OSHIMA Toshio Grad. Sch. of Math. Sci., The University of Tokyo, Prof., 大学院・数理科学研究科, 教授 (50011721)
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| Project Period (FY) |
1998 – 1999
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| Keywords | real reductive Lie groups / Unitary representations / degenerate series / derived functor modules |
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| Research Abstract |
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.
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