Cherednik Algebras and Affine Lie Algebras

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Representation Theory was born about 120 years ago by blending ideas from Algebra, Analysis, Geometry and Number Theory, with motivations coming from the then contemporary Physics. It keeps a central position in Mathematics to this day, and has found numerous applications in Biology, Chemistry, Information Technology and especially in Theoretical Physics. For instance, it was Representation Theory that gave mathematical foundations to the celebrated prediction made in 1964 by P. Higgs (Nobel Prize 2013) and confirmed by CERN experimenters in 2012. Reciprocally, Physics also remains a major source of inspiration for Representation Theory. This is particularly relevant to the present project, which is aimed to study certain algebraic objects emerged from the Theory of Integrable Systems and from the Quantum Field Theory. These algebraic objects were discovered around 1992 by I. Cherednik, who then used them to prove the famous conjectures of I. Macdonald on multivariate orthogonal polynomials. Since then these algebras have become the focus of attention of many world leading mathematicians, and found striking applications in Algebraic Geometry, Combinatorics and Mathematical Physics. We aim to advance the theory of these algebras by using the methods coming from the theory of affine Lie algebras.