Multiple flag varieties and exotic nilpotent varieties

2022 Fiscal Year Final Research Report

• Project/Area Number: 16K05070
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Aoyama Gakuin University
• Principal Investigator: Nishiyama Kyo 青山学院大学, 理工学部, 教授 (70183085)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Keywords: 旗多様体 (Flag varieties) / 二重旗多様体 / 対称部分群 / 組合せ論 (Combinatorics) / スタインバーグ写像 / ヘッケ環 (Hecke algebras) / 冪零多様体 / RSK対応

Outline of Final Research Achievements

Flag varieties are compact homogeneous spaces acted upon by a reductive Lie group, providing a broad field not only for the representation theory of Lie groups but also for the theories of geometry and algebra. In this research, we considered the product of flag varieties (double flag varieties) and studied the action of symmetric subgroups. Double flag varieties for which the orbits of symmetric subgroups are finite are called finite type. Various results were obtained in this research, including combinatorial aspects related to orbits on finite-type double flag varieties, representations of Hecke algebras, mappings to nilpotent varieties using moment maps, and the correspondence between orbits on double flag varieties and nilpotent orbits, known as the Steinberg correspondence, which is described combinatorially using a generalization of the RSK correspondence.

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

旗多様体は簡約リー群の作用するコンパクトな等質空間であって，リー群の表現論を始め，幾何学や代数学の理論に対しても幅広い舞台を提供している．本研究では旗多様体の直積（二重旗多様体）を考えて，対称部分群の作用を研究した．対称部分群の軌道が有限個であるような二重旗多様体は有限型と呼ばれる．有限型の二重旗多様体上の軌道にまつわる組合せ論やヘッケ環の表現，モーメント写像を用いた冪零多様体への写像，そしてスタインバーグ写像と呼ばれる二重旗多様体上の軌道と冪零軌道の対応，その組合せ論的記述であるRSK対応など様々な成果が本研究で得られた．