Mathematical Sciences: Algebraic Transformation Groups

9701200摘要：本项目由两个松散连接的部分组成。第一个是研究多重自由表示上的不变微分算子和球函数。人们获得了卡普里型等式，它导致了一种新的对称多项式。第二部分研究了Hamilton流形上的Weyl群。这可能导致一个关于多重自由哈密顿流形分类的Delzant猜想的证明。进一步的目标是开始系统的研究哈密尔顿代数簇。该项目的背景是研究像球面或双曲空间这样的高度对称空间。众所周知，这些空间的几何学的重要部分是由内部对称的某个有限群描述的，称为小外尔群。该项目的目的是调查两个不同方面的小外尔集团。首先是它的影响不变的微分算子，在某些情况下导致美丽的身份推广那些发现卡佩利在世纪。第二个目的是研究Hamilton流形的小Weyl群。这些流形自然出现，如果一个正式的对称性的经典力学系统，如顶部或行星系统。

9701200Abstract: This project consists of two loosely connected parts. The first is to study invariant differential operators and spherical functions on multiplicity free representations. One obtains Capelli-type identities which lead to a new kind of symmetric polynomials. The second part is to study Weyl groups for Hamiltonian manifolds. This might lead to a proof of the Delzant conjecture concerning the classification of multiplicity free Hamiltonian manifolds. A further goal is to begin a systematic study of Hamiltonian algebraic varieties.-The background of the project is the study of highly symmetric spaces like spheres or hyperbolic spaces. It is known that a significant part of the geometry of these spaces is described by a certain finite group of internal symmetries, called the little Weyl group. The aim of the project is to investigate two different aspects of the little Weyl group. The first is its influence on invariant differential operators which in certain cases lead to beautiful identities generalizing those found by Capelli in the 19th century. The second aim is to study the little Weyl group of Hamiltonian manifolds. These manifolds arise naturally if one formalizes symmetry of a classical mechanical system, like a top or a planetary system.