Studies on discrete series representations and the theory of automorphic forms

辛塔尼对西格尔尖形空间维数公式的研究[j]。东京，22(1975),25-65)非常令人印象深刻，我们必须了解Shinta美丽公式背后的一般机制，这是我们研究球函数的基本动机，以及它的傅里叶变换，半简单实数李群的离散级数表示。基本问题是1)估计矩阵系数以证明泊松求和公式，2)确定球面函数的傅里叶变换的非零集。在全纯离散级数的情况下，最简单的情况是10显式处理，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。这种表述将非零集与某些预齐次向量空间的开轨道的实轨道的连通分量联系起来，这些实轨道是由我们考虑傅里叶变换时用到的更多的代谢子群产生的。这种酉表示理论与预齐次向量空间理论之间的关系是一种新的关系，它给我们带来了一些有趣的问题。我们现在正处于研究的起点，我们有几种策略，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三重系统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一子群的限制分解。Jordan三重系统在第二种策略中也很有用，因为必须仔细选择全纯离散级数所限定的子群，而Jordan三重系统的语言给出了一个很好的子群选择。因此，我们必须深入研究约旦三元体系的群，以及这些群之间的关系。这些问题将是下一个由本奖助金所支持的计划的主题。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