Canonical bases in equivariant K-theory and their applications

• Project/Area Number: 17K14163
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: Hikita Tatsuyuki 京都大学, 数理解析研究所, 助教 (70793230)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 幾何学的表現論 / シンプレクティック特異点解消 / 標準基底 / 楕円コホモロジー / 楕円量子群 / シンプレクティック双対性 / 超幾何関数 / 錐的シンプレクティック特異点解消 / 表現論 / 代数幾何

Outline of Final Research Achievements

We defined a notion of canonical bases for equivariant K-theory of conical symplectic resolutions and formulated an analogue of Lusztig's conjecture for modular representation theory of semisimple Lie algebras. We explicitly calculated K-theoretic canonical bases for toric hyper-Kahler manifolds and proved a conjecture of Bezrukavnikov-Okounkov on the existence of real variation of stability conditions in this case. We also found that the equivariant lift of the central charge of the dual canonical bases are given by Euler type integral for GKZ hypergeometric functions over explicit cycles and proved that there exists a Koszul type t-structure on the derived category of equivariant coherent sheaves such that its heart has a graded highest weight category structure. We introduced elliptic analogue of caonical bases for toric hyper-Kahler manifolds and found a new duality of elliptic canonical bases under symplectic duality which cannot be seen at the K-theory level.

Academic Significance and Societal Importance of the Research Achievements

錐的シンプレクティック特異点解消の代数幾何や表現論に関して既存の予想の類似物を与えるだけでなく、様々な新しく興味深い現象を発見することができた。現在の技術では厳密に扱うことはまだ難しいが、ここで得られた結果は例えばシンプレクティック特異点解消の同変連接層の導来圏という代数幾何的な対象と、そのシンプレクティック双対のループ空間上の超局所偏屈層というシンプレクティック幾何的な対象が結びつくだろうというミラー対称性のような現象や、楕円標準基底の双対性と頂点作用素超代数の双対性に関係があるだろうといった様々な分野を結びつける現象を示唆している点が特に重要であると思われる。

Research Products

• [Presentation] Elliptic bar involutions for quiver varieties (2023)
  • Author(s): Tatsuyuki Hikita
  • Organizer: Representation theory and geometry of loop spaces
  • Int'l Joint Research / Invited
• [Presentation] Elliptic quantum group (2023)
  • Author(s): 疋田辰之
  • Organizer: 第6回Langlands and Harmonic Analysis
• [Presentation] K理論的標準基底とその楕円化 (2022)
  • Author(s): 疋田辰之
  • Organizer: 日本数学会2022年度年会
  • Invited
• [Presentation] Non-toric examples of elliptic canonical bases (2021)
  • Author(s): 疋田辰之
  • Organizer: Algebraic Lie Theory and Representation Theory
• [Presentation] Non-toric examples of elliptic canonical bases (2021)
  • Author(s): Tatsuyuki Hikita
  • Organizer: MS Seminar, Kavli IPMU
  • Invited
• [Presentation] Highest weight categories arising from equivariant coherent sheaves on toric hyper-Kahler manifolds (2020)
  • Author(s): 疋田辰之
  • Organizer: 阪大オンライン代数幾何学セミナー
• [Presentation] Elliptic canonical bases for toric hyper-Kahler manifolds (2020)
  • Author(s): 疋田辰之
  • Organizer: 東大京大代数幾何セミナー
• [Presentation] Elliptic canonical bases for hypertoric varieties (2019)
  • Author(s): Tatsuyuki Hikita
  • Organizer: Algebraic Lie Theory and Representation Theory 2019
• [Presentation] Elliptic and K-theoretic canonical bases for hypertoric varieties (2019)
  • Author(s): Tatsuyuki Hikita
  • Organizer: Representation Theory of Algebraic Groups and Quantum Groups
  • Int'l Joint Research / Invited
• [Presentation] Elliptic canonical bases for toric hyper-Kahler manifolds (2019)
  • Author(s): Tatsuyuki Hikita
  • Organizer: Geometric Representation Theory and Quantum Field Theories
  • Int'l Joint Research / Invited
• [Presentation] 標準基底と連接層 (2019)
  • Author(s): 疋田辰之
  • Organizer: 第2回数理新人セミナー
  • Invited
• [Presentation] Canonical bases and hypergeometric functions (2019)
  • Author(s): 疋田辰之
  • Organizer: 第4回Langlands and Harmonic Analysis
• [Presentation] Canonical bases in equivariant K-theory of conical symplectic resolutions (2017)
  • Author(s): 疋田辰之
  • Organizer: Algebraic Lie Theory and Representation Theory (ALTReT) 2017
  • Invited