Complete reducibility, geometric invariant theory, spherical buildings: a new approach to representations of algebraic groups

2021 Fiscal Year Final Research Report

• Project/Area Number: 19K14516
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Soka University
• Principal Investigator: Uchiyama Tomohiro 創価大学, 国際教養学部, 講師 (60822088)
• Project Period (FY): 2019-04-01 – 2022-03-31
• Keywords: 代数群

Outline of Final Research Achievements

In this project, we studied Serre's notion of complete reduciblity of subgroups of reductive algebraic groups using "geometric invariant theory" (a part of algebraic geometry) and "the theory of spherical buildings (highly symmetrical combinatorial objects). So far, the study of complete reducibility had been done by representation theoretic methods that were ad hoc, using different arguments depending on the "types" of reductive algebraic groups. In this research, I invented a unified method via geometric invariant theory and the theory of spherical buildings proving various results concerning complete reducibility in very short arguments.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

これまでタイプ別に分析されていた代数群の構造に幾何学的不変式論を用いた統一的理解とビルディングの理論を用いたトポロジー的解釈を与え、より構造を理解しやすくした。またこれまでほとんど分析されてこなかったより複雑な代数群のケース（体がperfectでないケース）に幾何学的不変式論を適用するとこれまで通りの結果が成り立つ場合と成り立たない場合がある事を具体例を使ってその原因とともに示した。この結果は特に数論への応用に対して重要であると考えられ、今後の発展がより期待される。