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Mod p representation theory and representation theory of Hecke algebras

Research Project

Project/Area Number 18H01107
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionThe 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
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥17,030,000 (Direct Cost: ¥13,100,000、Indirect Cost: ¥3,930,000)
Fiscal Year 2022: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2021: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2020: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2019: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2018: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Keywordsp進簡約群 / 既約表現 / 法p表現 / p進群 / p進バナッハ表現
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.

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.

Report

(6 results)
  • 2023 Final Research Report ( PDF )
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • Research Products

    (21 results)

All 2024 2023 2022 2020 2019 2018 Other

All Int'l Joint Research (3 results) Journal Article (8 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 8 results,  Open Access: 4 results) Presentation (9 results) (of which Int'l Joint Research: 3 results,  Invited: 6 results) Funded Workshop (1 results)

  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Institut de Mathematiques de Jussieu(フランス)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Toronto(カナダ)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] On the Irreducibility of p-Adic Banach Principal Series of p-Adic GL_32024

    • Author(s)
      Noriyuki Abe and Florian Herzig
    • Journal Title

      Vietnam Journal of Mathematics

      Volume: 52 Issue: 2 Pages: 451-478

    • DOI

      10.1007/s10013-023-00675-7

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Extension between simple modules of pro-pp-Iwahori Hecke algebras2023

    • Author(s)
      Noriyuki Abe
    • Journal Title

      J. Inst. Math. Jussieu

      Volume: 22 Issue: 6 Pages: 2804-2804

    • DOI

      10.1017/s1474748022000202

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On one-sided singular Soergel bimodules2023

    • Author(s)
      Noriyuki Abe
    • Journal Title

      J. Algebra

      Volume: 633 Pages: 722-753

    • DOI

      10.1016/j.jalgebra.2023.06.040

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Inverse Satake isomorphism and change of weight2022

    • Author(s)
      Noriyuki Abe, Florian Herzig and Marie-France Vigneras
    • Journal Title

      Representation Theory

      Volume: 26 Issue: 9 Pages: 264-324

    • DOI

      10.1090/ert/594

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A comparison between pro-p-Iwahori Hecke modules and mod p representations,2019

    • Author(s)
      Noriyuki Abe
    • Journal Title

      Algebra & Number Theory

      Volume: 13 Issue: 8 Pages: 1959-1981

    • DOI

      10.2140/ant.2019.13.1959

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Modulo p parabolic induction of pro-p-Iwahori Hecke algebra2019

    • Author(s)
      Noriyuki Abe
    • Journal Title

      Journal fur die reine und angewandte Mathematik

      Volume: 749 Issue: 749 Pages: 1-64

    • DOI

      10.1515/crelle-2016-0043

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Involutions on pro-p-Iwahori Hecke algebras2019

    • Author(s)
      Noriyuki Abe
    • Journal Title

      Representation Theory

      Volume: 23 Issue: 2 Pages: 57-87

    • DOI

      10.1090/ert/521

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On pro-p-Iwahori invariants of R-representations of reductive p-adic groups2018

    • Author(s)
      Noriyuki Abe, Guy Henniart and Marie-France Vigneras
    • Journal Title

      Representation Theory

      Volume: 22 Issue: 5 Pages: 119-159

    • DOI

      10.1090/ert/518

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] On Soergel bimodules2020

    • Author(s)
      Noriyuki Abe
    • Organizer
      Geometry and representation theory
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Soergel bimodules2020

    • Author(s)
      阿部紀行
    • Organizer
      第15回代数・解析・幾何学セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] On Soergel bimodules2019

    • Author(s)
      阿部紀行
    • Organizer
      2019年度RIMS共同研究(公開型) 「表現論とその周辺分野の進展」
    • Related Report
      2019 Annual Research Report
  • [Presentation] A Hecke action on G1_T-modules2019

    • Author(s)
      Noriyuki Abe
    • Organizer
      Modular Representation Theory
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Soergel bimodules2019

    • Author(s)
      阿部紀行
    • Organizer
      Arithmetic Geometry and Representation Theory
    • Related Report
      2019 Annual Research Report
  • [Presentation] Mod p representations and pro-p-Iwahori Hecke algebra2019

    • Author(s)
      Noriyuki Abe
    • Organizer
      Number / Representation Theory seminar(トロント大学)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Mod p representations and pro-p-Iwahori Hecke algebra2019

    • Author(s)
      阿部紀行
    • Organizer
      九大代数学セミナー(九州大学)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] p進代数群の法p表現とHecke環2018

    • Author(s)
      阿部紀行
    • Organizer
      代数学コロキウム(東京大学)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Mod p representations and pro-p Iwahori Hecke algebras2018

    • Author(s)
      Noriyuki Abe
    • Organizer
      2018 ICM satellite conference on Automorphic Forms, Galois Representations and L-functions
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Workshop on Shimura varieties, representation theory and related topics, 20192019

    • Related Report
      2019 Annual Research Report

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Published: 2018-04-23   Modified: 2025-01-30  

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