Studies on discrete series representations and the theory of automorphic forms

Shintani对Siegel尖点形式空间的维度公式(J.Fac.Science Univ.Tokyo，22(1975)，25-65)的研究是令人印象深刻的，我们必须了解Shinta美丽公式背后的一般机制，这是我们研究半单实Lie群离散级数表示的球函数及其傅立叶变换的基本动机。其基本问题是：1)为证明泊松求和公式而进行的矩阵系数估计，2)球函数傅里叶变换非零集的确定。在全纯离散级数的情况下，对于十个显式处理是最简单的情况，这种情况提出了关于球函数的傅里叶变换的非零集的一般猜想陈述。该陈述将非零集与由PAR…产生的某些准齐次向量空间的开轨道的实轨的连通分量联系起来更多的抛物子群，我们正在考虑傅里叶变换。么正表示理论与预齐次向量空间理论之间的这种关系是全新的，给我们带来了几个有趣的问题。我们现在正处于研究的起点，我们有几个策略，通过这些策略，我们可以最终建立两个完全不同的数学分支之间的关系。我们的策略之一是使用Jordan三元系来描述离散级数表示的显式构造。另一种策略是研究全纯离散级数对某个子群的限制的分解。在第二种策略中，Jordan三系也是非常有用的，因为必须仔细选择全纯离散级数所限制的子群，并且用Jordan三系的语言给出了一个很好的子群选择。因此，我们必须深入研究来自Jordan三系的群，以及这些群之间的关系。这些问题将是这笔赠款支持的下一个项目的主题。较少

Shintani's studies on the dimension formula of the space of Siegel cusp forms (J.Fac.Sci.Univ.Tokyo, 22(1975),25-65) 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 semi-simple 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 non-zero 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 non-zero set of the Fourier transform of spherical functions. That statement relates the non-zero set with the connected component of the real orbits of the open orbit of certain pre-homogeneous 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 pre-homogeneous 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