Explicit formulas of $p$-adic spherical functions and their applications

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540031
• 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)
• Co-Investigator(Renkei-kenkyūsha): SATO FUMIHIRO 立教大学, 理学部, 名誉教授 (20120884) ; KOMORI YASUSHI 立教大学, 理学部, 准教授 (80343200)
• Research Collaborator: Rubenthaler Hubert Strasbourg University, Institute of Mathematics, Professor ; Boecherer Siegfried Mannheim University, Institute of Mathematics, Professor
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: p進球関数 / ユニタリ・エルミート行列 / ユニタリ群 / ｐ進等質空間 / ヘッケ環 / マクドナルド多項式

Outline of Final Research Achievements

We have investigated the spaces of unitary-hermitian matrices on the basis of spherical functions as $p$-adic homogeneous spaces. We may apply a general expression formula of spherical functions which the researcher got before. The present groups have different root systems according to the parity of the size of matrices, and the Cartan decomposition of the spaces have different shapes according to the residual characteristic of the base field. We have studied at first the odd residual and even-size space, then the other cases. Finally we have a unified description for the results.

Free Research Field

数論