The construction of new research foundation for automorphic forms based on Fourier expansions in non-abelian directions

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03431
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Waseda University
• Principal Investigator: Narita Hiro-aki 早稲田大学, 理工学術院, 教授 (70433315)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: Fourier-Jacobi展開 / 次数２の斜交群 / 例外群G2 / Heisenberg群の表現論 / Jacobi群の表現論 / 一般化Eichler-Zagier対応 / 退化指標のWhittaker関数 / Fourier-Jacobi型球関数

Outline of Final Research Achievements

Automorphic forms, which are the target of this research, have rich symmetry called ``automorphy''. This symmetry includes the ``periodicity'', which trigonometric functions satisfy as is well known. As a series expansion explaining this property, there is a notion of the Fourier expansion. An achievement of this research is the establishment of a general theory of Fourier-Jacobi expansion for cusp forms on the symplectic group of degree two, which respects the periodicity induced by the non-abelian group action of the Heisenberg group. We also provide an idea of a proof for such expansion of cusp forms on the exceptional group of type G2.

Free Research Field

保型形式の整数論

Academic Significance and Societal Importance of the Research Achievements

三角関数等の有する周期性はユークリッド空間の座標の平行移動で説明されるが、保型形式が持つ周期性は一般にこれでは説明されない「非可換な周期性」を有することが多い。既存研究ではこの非可換周期性が持つ困難を「可換な周期性」と言えるユークリッド空間の平行移動に落とし込む操作をすることが多いが、本研究では「非可換な周期性」の持つ困難にヤコビ群の表現論などを駆使するなどして立ち向かい既存研究にない理論を打ち立てたことに価値がある。