Quiver varieties and Cherednik algebras

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.