| Project/Area Number |
15K17519
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
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| Research Institution | Sophia University |
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Principal Investigator |
NAKASUJI Maki 上智大学, 理工学部, 准教授 (30609871)
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| Research Collaborator |
BUMP Daniel
NARUSE Hiroshi
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | Whittaker関数 / Hecke環 / Casselman基底 / Casselman問題 / Kazhdan-Lusztig多項式 / Schur多重ゼータ関数 / 行列式表示 / Iwahori Hecke環 / Iwahori Whittaker関数 / Yang-Baxter基底 |
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| Outline of Final Research Achievements |
(1) Casselman basis is a basis vectors of the Iwahori Whittaker functions. Casselman problem is to get a closed formula for the change of basis between the Casselman basis and the natural basis. We obtained the solution to this problem by using the Yang-Baxter basis of Hecke algebra (joint work with H. Naruse).
(2) We introduced the deformation of the Kazhdan-Lusztig R-polynomial. As an application of it, under some assumption we proved Bump-Nakasuji conjecture which is the closed formula for the transition matrices between the deformation of the Casselman basis and natural basis, and obtained certain new functional equations for this transition matrices (joint work with D. Bump).
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| Academic Significance and Societal Importance of the Research Achievements |
Whittaker関数と組合せ論的表現論の連携は，2013 年頃から始まった新しい取り組みである.それぞれの理論には，Demazure 作用素や Bruhat 順序などの共通の道具があることは確認しているが，実際に領域間の関係を持ち込む研究は少ない.これに対し本研究は，Whittaker 関数の問題に対して，シューベルトカルキュラスやKazhdan-Lusztig多項式といった組合せ論的表現論を持込み問題の解決を試みた. 本研究は理論間の関係構築に貢献し，連携をより確実なものとしたと言える. また，これにより今後の解析数論およびその周辺の性質の解明に対する新しい展望を開くことが期待できる.
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