Mathematical Sciences: Algebraic Transformation Groups

摘要：本工程由两个松散连接的部分组成。首先是研究无多重表示下的不变微分算子和球函数。得到了capelli型恒等式，得到了一类新的对称多项式。第二部分是研究哈密顿流形的Weyl群。这可能导致关于无多重性哈密顿流形分类的Delzant猜想的证明。进一步的目标是开始对哈密顿代数变体进行系统的研究。项目的背景是研究高度对称的空间，如球体或双曲空间。众所周知，这些空间的很大一部分几何是由一个有限的内部对称群来描述的，这个群被称为小Weyl群。该项目的目的是调查小Weyl集团的两个不同方面。首先是它对不变微分算子的影响，在某些情况下，不变微分算子会产生漂亮的恒等式，这些恒等式推广了19世纪Capelli发现的恒等式。第二个目标是研究哈密顿流形的Weyl群。这些流形如果形式化一个经典机械系统的对称性，比如陀螺或行星系统，就会自然出现。

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.