2023 Fiscal Year Final Research Report
Study on irreducible representations of hyperspecial compact group and its applications
| Project/Area Number |
16K05053
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Miyagi University of Education |
|---|
Principal Investigator |
Takase Koichi 宮城教育大学, 教育学部, 特任教授 (60197093)
|
|---|
| Project Period (FY) |
2016-04-01 – 2024-03-31
|
| Keywords | Weil 表現 / 超尖点的既約表現 / Jordan 三重系 / Jordan 代数 / 正則離散系列表現 / 球関数 / Laplace 変換 |
|---|
| Outline of Final Research Achievements |
1)A natural parametrization of the regular irreducible complex linear representations of the finite group defined as a finite reduction of a group scheme defined over the integer ring of a non-Archimedian local foiled is given. The results are presented by Regular irreducible representations of classical groups over finite quotient rings (Pacific Journal of Mathematics (2021) 221-256)
2) As an application of the results of 1), certain supercuspidal representations of the symplectic group and the special linear group over non-Archimedian local field are explicitely constructed, and the formal degree conjecture and the root number conjecture are verified.THe results are presentated by On certain supercuspidal representations of symplectic groups associated with tamely ramified extensions : the formal degree conjecture and the root number conjecture (arXiv:2109.07124).
|
| Free Research Field |
局所体上の代数群の表現論
|
| Academic Significance and Societal Importance of the Research Achievements |
局所体の整数環上定義された群スキームから reduction により生じる有限群の既約表現を決定することは，有限群の一般論から興味深いばかりではなく，局所体上の代数群の超尖点的既約表現を具体的に構成する際に非常に有効に用いることが出来るので，極めて重要な研究課題である．本研究では，有限群の既約表現の中でも generic な位置にある regular な既約表現の完全なパラメータ付けを与えることに成功した．更にその応用として，斜交群と特殊線形群の超尖点的既約表現を具体的に構成して，「形式的次数予想」と「ルートナンバー予想」が成り立つことを確かめることができた．
|
