Arithmetic study of automorphic forms of many variables by various method

2014 Fiscal Year Final Research Report

• Project/Area Number: 23244003
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: ODA Takayuki 東京大学, 数理(科)学研究科(研究院), 教授 (10109415)
• Co-Investigator(Renkei-kenkyūsha): HIRONAKA Yumiko 早稲田大学, 教育学部, 教授 (10153652) ; WAKATSUKI Satoshi 金沢大学, 数物系, 准教授 (10432121) ; KOSEKI Harutaka 三重大学, 教育学部, 教授 (60234770) ; HAYATA Takahiro 山形大学, 大学院理工学研究科, 准教授 (50312757) ; TSUZUKI Masao 上智大学, 理工学部, 准教授 (80296946) ; HIRANO Miki 愛媛大学, 大学院理工学研究科, 教授 (80314946) ; GON Yasuro 九州大学, 数理学研究院, 准教授 (30302508) ; ISHI Taku 成蹊大学, 理工学部, 准教授 (60406650)
• Project Period (FY): 2011-04-01 – 2015-03-31
• Keywords: Automorphic forms / spherical functions / Green functions / Whittaker function / modular forms

Outline of Final Research Achievements

We obtained some fundamental results on the integral expressions and power series expressions of the A-radial parts of either Whittaker functions or spherical functions for the standard representations (i.e principal series and/or discrete series representations) of the Lie groups, GL(n,R), Sp(2,R) and SU(3,1).The formulas of Whittaker functions of non-spherical principal series put a period on the research history beginning from the studies of D. Bump and others, and we can expect various applications of this result (this is a joint works with Taku Ishii of Seikei Univ.). We obtained an explicit formulas of the matrix coefficients of the large discrete series of the Lie groups SU(2,1), SU(3,1) (joint wrok with T.Hayata, H. Koseki, and T. Miyazaki).This result gives a suggestion for study of the reproducing kernels. We push forward the investigation oh the cell-decomposition of Siegel-Gottschling fundamental domain of genus 2 (the first paper was a joint paper with T. Hayata).

Free Research Field

Number Theory