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.
