Harmonic Analysis on p-adic homogeneous spaces based on spherical functions

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05081
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Waseda University
• Principal Investigator: HIRONAKA Yumiko 早稲田大学, 教育・総合科学学術院, 教授 (10153652)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: p進等質空間 / 球関数 / quaternion hermitian / 局所密度 / Macdonald多項式 / ヘッケ環

Outline of Final Research Achievements

In this period, we have studied the spherical functions on the space of unitary hermitain forms with even residual characteristic and on the space of quaternion hermitian forms with odd residual characteristic, as interesting $p$-adic homogeneous spaces from the perspective of Number Theory. For both cases, we have constructed the typical spherical functions from relative invariants, and given those functional equations with respect to the Weyl groups, then obtained the explicit formulas by using our previous result on general expression formulas of spherical functions. In the main terms of the spherical functions there appear a series of Weyl-invariant polynomials and play an important role for the harmonic analysis on the spaces through the spherical transform.

Free Research Field

Number Theory

Academic Significance and Societal Importance of the Research Achievements

$p$進等質空間として興味ある半双線形形式の空間について球関数に基づく詳細な研究を行っている．球関数の明示式にはワイル群に対応するMacdonald(直交)多項式系の特殊化や類似が現れ，抽象的な直交多項式系を具現する等質空間を与えたことにもなる．
また，同時に現れる組み合わせ論的な量も興味深い．また，球関数の使いやすい明示式に基づく空間の調和解析的手法の研究により，基底問題や早退積公式などの大局的保形形式とも緊密に結びつく．