A constructive study of real analytic automorphic forms using the branching rule of representations of reductive Lie groups as leverage

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05172
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: MORIYAMA Tomonori 大阪大学, 大学院理学研究科, 准教授 (80384171)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 保型形式 / 表現論 / テータ級数 / フーリエ係数

Outline of Final Research Achievements

Although there are many known methods of constructing concrete examples of real analytic automorphic forms, few of them stand up to precise study compared to holomorphic automorphic forms. The purpose of this project was to remedy this situation by using the branching rules of representations of reductive Lie groups. As a first and important step, we succeeded in constructing an integral expression for generalized Whittaker functions on real symplectic group of rank two, which generate certain generalized principal series representations. This integral expression is given by a double inverse Mellin transform.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

今回得られた一般化Whittaker函数の積分表示式は、パラメータに関する挙動が調べやすく保型L函数への応用上便利であろう。また、多くの困難が予想される一般化Whittaker函数を用いたポアンカレ級数を用いた実解析的ジーゲル保型形式の構成にも役立てられると思われる。