2019 Fiscal Year Final Research Report
Analysis on the structure of standard Whittaker modules
| Project/Area Number |
16K05071
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Aoyama Gakuin University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 標準 Whittaker 加群 / リー群の表現の組成列 / Whittaker 模型 / 確定特異点型偏微分方程式系の境界値問題 |
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| Outline of Final Research Achievements |
In the representation theory of groups, it is one of the basic problems to define standard representations and determine the structure of them.
In this research, I worked on this problem about the standard Whittaker (g,K)- modules, which I had defined. The results are as follows.(1) The socle filtrations of the standard Whittaker (g, K)-modules of Sp(2,R) are completely determined. (2) In the case of split groups, I proved that these modules are stable under the translation functors and I determined the global characters of the standard Whittaker (g, K)-modules. I also tried to show the self duality conjecture, which is one of the main objects of this research. Though important observations are obtained, this conjecture is still unproved.
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| Free Research Field |
リー群の表現論
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| Academic Significance and Societal Importance of the Research Achievements |
群や環の表現論において，標準的な表現を構成し，その構造解析を行うことは基本的かつ重要な問題である．本研究では，Whittaker 関数の空間という誘導表現から構成された標準 Whittaker (g,K)-加群に対して，この問題に取り組んだ．研究を続ける過程で，この加群はある圏における入射加群であることが認識され，この事実を使うことで，群が split の場合には，translation での安定性や大域指標の決定，即ち組成因子問題を解決することができた．当初は解析的な問題と思われていたが，全く異なる代数的な手法で研究が進展したことは興味深いと考えている．
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