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Geometric Representation theory of reductive group

Research Project

Project/Area Number 11440011
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTokyo Institute of Technology (2001)
Kyushu University (1999-2000)

Principal Investigator

OCHIAI Hiroyuki  Graduate school of Science and Engineering, Tokyo Institute of Technology, AP, 大学院・理工学研究科, 助教授 (90214163)

Co-Investigator(Kenkyū-buntansha) KANEKO Masanobu  Kyushu University, Mathematics, P, 大学院・数理学研究院, 教授 (70202017)
OSHIMA Toshio  The University Tokyo, Mathematical Science, P, 大学院・数理科学研究科, 教授 (50011721)
WAKIMOTO Minoru  Kyushu University, Mathematics, P, 大学院・数理学研究院, 教授 (00028218)
KONNO Takuya  Kyushu University, Mathematics, AP, 大学院・数理学研究院, 助教授 (00274431)
SHIMENO Nobukazu  Okayama Science University, Science, AP, 理学部, 助教授 (60254140)
村田 實  東京工業大学, 大学院・理工学研究科, 教授 (50087079)
村上 斉  東京工業大学, 大学院・理工学研究科, 助教授 (70192771)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥8,100,000 (Direct Cost: ¥8,100,000)
Fiscal Year 2001: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2000: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 1999: ¥3,200,000 (Direct Cost: ¥3,200,000)
Keywordssemisimple / highest weight / degree / theta correspondence / nilpotent orbit / associated variety / prehomogeneous / Kazhdan-Lusztig / 半単純リー郡 / テータ対応 / 最高ウエイト表現 / Giambelliの公式 / 重複度自由 / ベキ零軌道 / Selberg 積分 / Kazhdan-Lusztig予想 / ガンマ因子 / 保型形式
Research Abstract

The Bernstein degree, associated cycles and isotropy representation for a unitary highest weight representation of reductive Lie groups are important family of geometric invariants. For the symplectic groups, Ochiai with K. Nishiyama determines the Bernstein degree. For orthogonal and unitary groups, Nishiyama, Ochiai and K. Taniguchi determines the Bernstein degree and associated cycles. The general case including exceptional groups are done by Ochiai and Shohei Kato. This results are generalized to describe the degree of spherical orbits of a family of multiplicity-free representations by Ochiai and Kato. This is a new application of the Selberg-type integral.
A representation is considered as a quantization of a function space. Oshima introduces the notion of homogenized enveloping algebra, in order to deal with universal enveloping algebras and coordinate rings simultaneously. Using the Capelli operator, which is a quantization of minors, Oshima describes the annihilator ideals of generalized Verma modules of scalar type. Related to this work, Ochiai describes the difference of the centers of the enveloping algebras and the rings of invariant differential operators. Based on the work above, Oshima with Shimeno characterize the image of the Poisson transform associated to non-minimal parabolic.
Wakimoto discuss the modular invariance of characters of various affine algebras. Kaneko investigates the modular function *j from the geometry of singular moduli and supersingular elliptic curves over finite fields. Ochiai also discuss the quasi-modularity of the generating functions of an elliptic curve with a specified type of ramification indeces. Konno considers the unipotent representations of reductive groups over local fields, especially gives the description of CAP representations, which is a special subclass of unipotent representation, for low-rank groups.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (48 results)

All Other

All Publications (48 results)

  • [Publications] Hiroyuki Ochiai: "Non-commutative harmonic oscillators and Fuchsian ordinary differential equations"Communications in Mathematical Physics. 217. 357-373 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi: "Bernstein degree and associated cycles of Harish-Chandra modules Bernstein degree and associated cycles of Harish-Chandra modules"Asterisque, Societe Mathematique de France. 273. 13-80 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shohei Kato, Hiroyuki Ochiai: "The degree of orbits of multiplicity-free actions"Asterisque, Societe Mathematique de France. 273. 139-158 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ishihara, H.Ochiai, Y.Takegahara, T.Yoshida: "On the power of a prime $p$ dividing the number of solutions of x^p=1 in a symmetric group"Annals of Combinatorics. 5. 197-210 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Victor Kac, Minoru Wakimoto: "Integrable highest weight modules over affine superalgebras and Appell's function"Communications in Mathematical Physics. 215. 631-682 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hitoshi Murakami, Jun Murakami: "The colored Jones polynomials and the simplicial volume of a knot"Acta Math.. 186. 85-104 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Nishiyama, Hiroyuki Ochiai: "Bernstein degree of singular unitary highest weight representations of the metaplectic group"Proceedings of Japan Academy. 75. 9-11 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroyuki Ochiai: "A p-adic property of Taylor series of exp(x+x^p/p)"Hokkaido Math. J.. 28. 71-85 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroyuki Ochiai: "Classification of commuting differential operators with two variables"Yang-Baxter systems, non-linear models and their applications. 183-187 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Kaneko, N.Todaka: "Hypergeometric modular forms and supersingular elliptic curves"CRM Proceedings and Lecture Notes. 30. 79-83 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N.Shimeno: "Boundary value problems for various boundaries of Hermitian symmetric spaces"Journal of Functional Analysis. 170. 265-285 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Konno: "Twisted endoscopy and the generic packet conjecture"Israel Journal of Mathematics. 129. 253-290 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Minoru Wakimoto: "World Scientific Publishing Company"Lectures on Infinite-Dimensional Lie Algebra,. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Ochiai: "A p-adic property of Taylor series of exp(x+x^p/p)"Hokkaido Math. J.. 28. 71-85 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Nishiyama and H. Ochiai: "Bernstein degree of singular unitary highest weight representations of the metaplectic group"Proc. Japan Acad.. 75 Ser. A. 9-11 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Ochiai: "Classification of commuting differential operators with two variables, in Yang-Baxter systems, non-linear models and their applications"World Sci.. 183-187 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Ochiai: "Non-commutative harmonic, oscillators and Fuchsian ordinary differential equations, Kyushu Univ. preprint series 1998-18"Comm. Math. Phys.. 217, no. 2. 357-373 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Nishiyama, Hiroyuki Ochiai and K. Taniguchi: "Bernstein degree and associated cycles of Harish-Chandra modules (with K. Nishiyama and K. Taniguchi) Bernstein degree and associated cycles of Harish-Chandra modules Hermitian symmetric case"Asterisque. 273. 13-80 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shohei Kato and H. Ochiai: "The degree of orbits of multiplicity-free actions (with Shohei Kato)"Asterisque. 273. 139-158 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Ishihara, H. Ochiai, Y. Takegahara, T. Yoshida: "On the power of a prime p dividing the number of solutions x^p = 1 in a symmetric group"Annals of Combinatorics. 5. 197-210 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] V G. Kac and M. Wakimoto: "Integrable highest modules over affine superalgebras and Appell's function"Comm. Math. Phys.. 215 (Joint work with V. G. Kac) (to appear). 631-682 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Kaneko and N. Todaka: "Hypergeometric modular forms and super-singular elliptic curves, Proceedings on Moonshine and related topics (J. Mckay and A. Sebbar ed.)"CRM Proceedings and Lecture Notes. vol. 30. 79-83 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N. Shimeno: "Boundary value problems for various boundaries of Hermitian symmetric spaces"J. of Funct. Anal.. 170. 265-285 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Konno: "Twisted endoscopy and the generic packet conjecture"Israel J. Math.. 129. 253-290 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Murakami and J. Murakami: "The colored Jones polynomials and the simplicial volume of a knot"Acta Math.. 186. 85-104 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroyuki Ochiai: "Non-commutative harmonic oscillators and Fuchsian ordinary differential equations"Communications in Mathematical Physics. 217. 357-373 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi: "Bernstein degree and associated cycles of Harish-Chandra modules Bernstein degree and associated cycles of Harish-Chandra modules"Asterisque, Societe Mathematique de France. 273. 13-80 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shohei Kato, Hiroyuki Ochiai: "The degree of orbits of multiplicity-free actions"Asterisque, Societe Mathematique de France. 273. 139-158 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Ishihara, H.Ochiai, Y.Takegahara, T.Yoshida: "On the power of a prime $p$ dividing the number of solutions of x^p=1 in a symmetric group"Annals of Combinatorics. 5. 197-210 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Victor Kac, Minoru Wakimoto: "Integrable highest weight modules over affine superalgebras and Appell's function"Communications in Mathematical Physics. 215. 631-682 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hitoshi Murakami, Jun Murakami: "The colored Jones polynomials and the simplicial volume of a knot"Acta Math.. 186. 85-104 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masanobu Kaneko, Naoya Todaka: "Hypergeometric modular forms and supersingular elliptic curves"Proceedings on Moonshine and related topics(J. Mckay and A. Sebbar ed.), CRM Proceedings and Lecture Notes. 30. 79-83 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Minoru Wakimoto: "World Scientific Publishing Company"Lectures on Infinite-Dimensional Lie Algebra. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Ochiai: "Classification of commuting differential operators with two variables"Yang-Baxter systems, non-linear models and their application. 183-187 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Nishiyama,H.Ochiai,K.Taniguchi: "Bernstein degree and associated cycles of Harish-Chandra modules"Asterisque. (発表予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kato,H.0chiai: "The degree of orbits of multiplicity-free actions"Asterisque. (発表予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Ochiai: "Non-commutative harmonic oscillators and Fuchsian ordinary differential equations,"Communications in Mathematical Physics. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Wakimoto: "Non-commutative harmonic oscillators and Fuchsian ordinary differential equations,"Advanced Studies in Pure Math. "Analysis on Homogeneous Spaces and Representation Theory of Lie Groups". 26. 325-353 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.-J.Cheng and V.G.Kac,M.Wakimoto: "Extensions of Neveu-Schwarz conformal modules"Jour.Math.Phys.. 41. 2271-2294 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] V.Kac,M.Wakimoto: "Integrable highest weight modules over affine superalgebras and Appell's function"Communications in Mathematical Physics. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Kaneko: "The Akiyama-Tanigawa algorithm for Bernoulli numbers"Electronic J.of Integer Sequences. 3. (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Kaneko,N.Todaka: "Hypergeometric modular forms and supersingular elliptic curves"The proceedings of the Moonshine conference in Montr\'ea1. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Kaneko,K.Tachibana: "When is a polygonal pyramid number again polygonal?"Rockey Mountain Journal. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] N.Shimeno: "An analogue of Hardy's theorem on the Poincare disk"Bull.of Okayama Univ.of Science. 36A. 7-10 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] N.Shimeno: "A note on the uncertainty principle for the Dunkl transform"J.Math.Sci.Univ.Tokyo. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Konno: "Twisted endoscopy and the generic packet conjecture,"Israel Journal of Mathematics. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Nishiyama,H.Ochiai: "Bernstein degree of singular unitary highest weight represed&tions metaplectic group"Proc.Japan Acad.. 75. 9-11 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "A p-adic property of Taylor series exp(X+XP/P)"Hokkaido Math.J.. 28. 71-85 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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