Noncommutative Deligne-Lusztig theory

• Project/Area Number: 17K18722
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Algebra, Geometry, and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: Imai Naoki 東京大学, 大学院数理科学研究科, 准教授 (90597775)
• Project Period (FY): 2017-06-30 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥6,240,000 (Direct Cost: ¥4,800,000、Indirect Cost: ¥1,440,000); Fiscal Year 2019: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2018: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2017: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• Keywords: Deligne-Lusztig 構成 / 代数学

Outline of Final Research Achievements

By considering an analogue of Deligne-Lusztig theory, we constructed representations which we are interested in from a number theoretic view point. More concretely, we constructed a Weil representation for a finite unitary group and its Shintani lift geometrically. Further, using an Frobenius action, we constructed and studied the Howe correspondence for a symplectic group in a special case. We studied also mod ell Weil representations and mod ell Howe correspondences.

Academic Significance and Societal Importance of the Research Achievements

数論的に興味深い表現を幾何学的に構成することができた．またその構成を用いて，新谷リフトのような関手性を幾何学的に実現したり，表現に関する性質を幾何的な手法を用いて調べることができた．さらに幾何学的に実現できていることを用いて mod ell 係数における類似の対象を自然に構成し，それらについても幾何的に調べて新しい結果が得られた．modular 表現は数論においても重要な役割を果たすため，今後の応用も期待でき，学術的意義があると考えられる．

Research Products

• [Journal Article] Geometric realization of the local Langlands correspondence for representations of conductor three (2020)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: Publ. Res. Inst. Math. Sci. Volume: N/A
  • Peer Reviewed
• [Journal Article] Affinoids in the Lubin-Tate perfectoid space and simple supercuspidal representations I: tame case (2019)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: International Mathematics Research Notices Volume: 印刷中 Pages: 8251-8291
  • DOI: 10.1093/imrn/rny229
  • Peer Reviewed / Open Access
• [Presentation] Convolution morphisms and Kottwitz conjecture (2020)
  • Author(s): Naoki Imai
  • Organizer: Caltech number theory seminar
  • Int'l Joint Research / Invited
• [Presentation] Deligne-Lusztig stack (2019)
  • Author(s): Naoki Imai
  • Organizer: International Conference on Arithmetic Geometry, In honor of Michael Rapoport's 71st birthday
  • Int'l Joint Research / Invited
• [Presentation] Langlands functoriality in the geometrization of the local Langlands correspondence (2019)
  • Author(s): Naoki Imai
  • Organizer: New Developments in Representation Theory of p-adic Groups
  • Int'l Joint Research / Invited
• [Presentation] Convolution morphisms, geometric Satake equivalence and Kottwitz conjecture (2019)
  • Author(s): Naoki Imai
  • Organizer: The conference on the Legacy of Elie Cartan
  • Int'l Joint Research / Invited
• [Presentation] Geometric realization of Heisenberg-Weil representations for finite unitary groups (2018)
  • Author(s): Naoki Imai
  • Organizer: Tokyo-Lyon Satellite Conference in Number Theory
  • Int'l Joint Research / Invited