Mathematical Sciences: Algebraic Transformation Groups

摘要：本项目由两个松散连接的部分组成。第一个是研究无重数表示上的不变微分算子和球函数。得到了Capelli型恒等式，由此得到了一类新的对称多项式。第二部分研究了哈密顿流形上的Weyl群。这可能导致关于无重数哈密顿流形的分类的Delzant猜想的证明。另一个目标是开始系统地研究哈密尔顿代数簇。-该项目的背景是研究高度对称的空间，如球面或双曲空间。众所周知，这些空间的很大一部分几何是由一个内部对称的有限群来描述的，称为小Weyl群。该项目的目的是调查小Weyl群的两个不同方面。第一个是它对不变微分算符的影响，在某些情况下，不变微分算符导致了美丽的恒等式，推广了Capelli在19世纪发现的恒等式。第二个目的是研究哈密顿流形的小Weyl群。如果人们形式化一个经典机械系统的对称性，比如顶层或行星系统，这些流形就会自然产生。

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.