Twistor transform for indefinite Grassmannian manifolds and the theory of infinite-dimensional representations

• Project/Area Number: 23540073
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: Sekiguchi Hideko 東京大学, 数理(科)学研究科(研究院), 准教授 (50281134)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2015: ¥260,000 (Direct Cost: ¥200,000、Indirect Cost: ¥60,000); Fiscal Year 2014: ¥260,000 (Direct Cost: ¥200,000、Indirect Cost: ¥60,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥260,000 (Direct Cost: ¥200,000、Indirect Cost: ¥60,000); Fiscal Year 2011: ¥260,000 (Direct Cost: ¥200,000、Indirect Cost: ¥60,000)
• Keywords: ペンローズ変換 / ユニタリ表現 / 有界対称領域 / 表現の分岐則 / 複素多様体 / リー群 / グラスマン多様体 / 積分幾何 / エルミート対称空間

Outline of Final Research Achievements

The irreducible decomposition of scalar holomorphic discrete series representations when restricted to semisimple symmetric pairs (G,H) is explicitly known as the Kobayashi--Schmid rule. This was proved by Schmid for H compact and by Kobayashi for general H. More generally, Kobayashi's multiplicity-free theorem ('97) guarantees that the restriction π|H is multiplicity-free whenever (G,H) is a symmetric pair and π is an irreducible unitary highest weight module of scalar type. During the period of research, I studied the Penrose transform for indefinite Grassmannian manifolds, and as its application, obtained some branching laws of singular highest weight modules with respect to the pair (U(n,n), SO*(2n)). This gives an extension of the Kobayashi--Schmid formula to certain non-tempered unitary representations which are realized in Dolbeault cohomology groups over indefinite Grassmannian manifolds. The resulting branching rule is multiplicity-free and discretely decomposable.

Research Products

• [Journal Article] Radon--Penrose transform between symmetric spaces (2013)
  • Author(s): Hideko Sekiguchi
  • Journal Title: Contemporary Mathematics Volume: 598
  • Peer Reviewed
• [Journal Article] Branching rules of singular unitary representations with respect to symmetric pairs (A_{2n-1}, D_n) (2013)
  • Author(s): Hideko Sekiguchi
  • Journal Title: International Journal of Mathematics Volume: 24 Issue: 04 Pages: 1350011-1350011
  • DOI: 10.1142/s0129167x13500110
  • Peer Reviewed
• [Journal Article] Radon--Penrose transform between symmetric spaces (2013)
  • Author(s): H. Sekiguchi
  • Journal Title: to appear in Contemporary Mathematics Volume: -
  • Peer Reviewed
• [Journal Article] Branching rules of singular unitary representations with respect to symmetric pairs $(A_{2n-1}, D_n)$ (2013)
  • Author(s): H. Sekiguchi
  • Journal Title: to appear in International Journal of Mathematics Volume: -
  • Peer Reviewed
• [Journal Article] Branching rules of Dolbeault cohomology groups over indefinite Grassmannian manifolds (2011)
  • Author(s): Hideko Sekiguchi
  • Journal Title: Proceedings of the Japan Academy, Ser. A Mathematical Sciences Volume: 87
  • NAID: 40018844395
  • Peer Reviewed
• [Journal Article] Penrose transform for indefinite Grassmann manifolds (2011)
  • Author(s): Hideko Sekiguchi
  • Journal Title: International Journal of Mathematics Volume: 22
  • Peer Reviewed
• [Presentation] Penrose transform between symmetric spaces (2013)
  • Author(s): Hideko Sekiguchi
  • Organizer: The Asian Mathematical Conference(AMC2013)
  • Place of Presentation: BEXCO, Busan, Korea
  • Invited
• [Presentation] Penrose transform between symmetric spaces (2012)
  • Author(s): Hideko Sekiguchi
  • Organizer: 2012 Joint Mathematics Meetings, AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason(招待講演)
  • Place of Presentation: Boston (アメリカ合衆国)