Quiver varieties and Cherednik algebras

该项目属于表象理论、几何和数学物理领域。主要目的是研究箭形簇(及其适当的推广)与Cherednik和双重仿射Hecke代数(DAHAS)的表示理论的关系。在A型(与对称群有关)中已经得到了一些结果，但要将它们推广到其他类型是一项具有挑战性的任务。最有趣也是最棘手的病例之一是BC型大哈。预计它的不可约有限维表示的多样性(在经典水平Q=1)应该与星形箭图的乘性箭图种类有关，而主要项目目标之一将是建立这一点。BC类型的DAHA与其他主题(如簇代数、纽结理论和可积系统)之间有几个有趣的联系，因此该项目的一部分将致力于探索这些联系。该项目属于代数、几何与拓扑学和数学物理的研究领域，所有这些都在EPSRC的文件夹中得到了很好的代表。

The project belongs to the area of representation theory, geometry, and mathematical physics. The main goal is to study quiver varieties (and their suitable generalisations) in relation with the representation theory of Cherednik and Double Affine Hecke algebras (DAHAs). Some results have been obtained in type A (related to the symmetric group), but it is a challenging task to generalise them to other types. One of the most interesting and difficult cases is DAHA of BC-type. It is expected that the variety of its irreducible finite-dimensional representations (at the classical level q=1) should be related to multiplicative quiver varieties for a star-shaped quiver, and one of the main project goals will be to establish this. There are several intriguing connections between DAHAs of BC-type and other topics such as cluster algebras, knot theory, and integrable systems, so part of the project will be devoted to exploring those links. The project belongs to the research areas of Algebra, Geometry and Topology, and Mathematical Physics, all of which are well represented in the EPSRC portfolio.