Mod p representation theory and representation theory of Hecke algebras

2023 Fiscal Year Final Research Report

• Project/Area Number: 18H01107
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The University of Tokyo
• Principal Investigator: Abe Noriyuki 東京大学, 大学院数理科学研究科, 教授 (00553629)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: p進簡約群 / 既約表現 / 法p表現

Outline of Final Research Achievements

I studied the theory of mod p representations of p-adic groups by linking it to the representation theory of pro-p Iwahori-Hecke algebras with the goal of contributing to the mod p Langlands correspondence. While it was already known that there was a partial correspondence between the irreducible representations of the pro-p Iwahori-Hecke algebra and the irreducible mod p representations, I upgraded this correspondence to a categorical level.

Furthermore, with the aim of investigating the representation theory of compact open subgroups, I studied the theory of algebraic representations of reductive algebraic groups and provided a new realization of the Hecke category that describes the structure of algebraic representations. As a joint work with F. Herzig, I studied the irreducibility of principal series representations of p-adic Banach representations, which are as significant as mod p representations.

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

法p表現は単に既約表現自身だけではなくその圏論的性質も重要と思われる．それをプロp岩堀Hecke環と関連付けた成果は一定の価値があると思われる．Hecke圏の新しい実現はすでに簡約代数群の表現圏へのHecke作用の構成などへの応用を持ち，今後さらに活用されていくと期待している．p進Banach主系列表現の既約性はこれまで殆ど知られていなかった．新たな知見を持ち込んだ本研究は意義深いと考えている．