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

2015 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

非可換調和解析