2008 Fiscal Year Final Research Report Summary
Studies on discrete series representations and the theory of automorphic forms
Project/Area Number  17540005 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Miyagi University of Education 
Principal Investigator 
TAKASE Koichi Miyagi University of Education, Department of Mathematics, Professor (60197093)

CoInvestigator(Kenkyūbuntansha) 
SATO Fumihiro Rikkyo University, Department of Mathematics, Professor (20120884)
NISHIYAMA Kyo Aoyama Gakuin University, Department of Physics And Mathematics, Professor (70183085)
OCHIAI hiroyuki Kyushu University, Faculty of Mathematics, Professor (90214163)

Project Period (FY) 
2005 – 2007

Keywords  automorphic form / unitary representation / spherical function / prehomogeneous vector sac / Fourier transform / Jordan triple system 
Research Abstract 
Shintani's studies on the dimension formula of the space of Siegel cusp forms (J.Fac.Sci.Univ.Tokyo, 22(1975),2565) are quite impressive, and we must understand the general mechanism working behind the beautiful formula of Shinta's, and that is the fundamental motivation of our studies on the spherical function, and Fourier transform of it, of discrete series represe tations of semisimple real Lie group. The basic problems are 1) estimate of matrix coefficients in order to justify the Poisson summation formula, 2) the determination of the nonzero set of the Fourier transform of the spherical functions. In the case of holomorphic discrete series is the most easy case tfor ten explicit treatment, and that case suggests the general conjectural statements on the nonzero set of the Fourier transform of spherical functions. That statement relates the nonzero set with the connected component of the real orbits of the open orbit of certain prehomogeneous vector space arising from the par
… More
abolic subgroup with which we are considering the Fourier transform. Such kind of relation between the theory of unitary representation and the theory of prehomogeneous vector space is uite new, and give us several interesting problems. We are now at the starting point of the study, and we have several strategies by which we may finally establish the relation between two quite different branch of Mathematics. One of our strategy is to use Jordan triple system to describe the explicit construction of discrete series representaions. The other strategy is to study the decomposition of the restriction of taholomorphic discrete series to certain subgroup. Jordan triple system is also quite useful in the second strategy, because the subgroup to which the holomorphic discrete series is restricted must be carefully chosen, and a good choice of the subgroup is given by the language of Jordan triple system. So we must study intensively the groups coming from Jordan triple system, and also the relation between these groups. These problms will be the theme of the next project supported by this grant in aide. Less

Research Products
(5results)