Geometric methods in representation theory of rational Cherednik algebras.

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This proposal will investigate certain problems in representation theory, a major branch of algebra interacting strongly with geometry and mathematical physics. Pure mathematics aims to abstract and distil the essence of familiar concepts: in the case of symmetries this leads to the definition of a group, the collection of symmetries of a given object. However, the mathematical definition is far more axiomatic, to the point that the underlying object that the group is describing all but disappears. In these cases it is important to try to recover this object, or more specifically to find all objects whose symmetries give rise to the group in question. This is the motivating idea behind representation theory. Despite this seemingly abstract problem, representation theory is crucially important in many areas of science such as physics (e.g. string theory / mirror symmetry), chemistry (study of molecular vibrations) and computer science, as well as being central for mathematics.Rational Cherednik algebras, as introduced by Etingof and Ginzburg relate to, and build upon results in symplectic algebraic geometry, Lie theoretic and geometric representation theory, and algebraic combinatorics. In particular, they have already been used to prove very difficult results such as answering the question of existence of crepant resolutions for symplectic quotient singularities and solving combinatorial conjectures on the properties of certain rings of coinvariants related to Mark Haiman's n!-conjecture. These results illustrate the power of applying the techniques that exist in the representation theory of noncommutative algebras to solving hard problems in related areas of pure mathematics. There are two parts to this proposal. In the first part I plan to investigate the connection between rational Cherednik algebras at t=0 to affine Lie algebras at the critical level, thereby providing a way of using the powerful tools already developed in that area (such as the geometry of Opers) by Frenkel, Gaitsgory and others to gain a much better understanding of the representation theory of rational Cherednik algebras. It is also natural to expect that our understanding of rational Cherednik algebras will have many interesting applications in the study of affine Lie algebras at the critical level. One of the most celebrated results in the field of representation theory in the past 30 years has been the proof by Beilinson and Bernstein of the Kazhdan-Luzstig conjecture using the idea of localization. In the second part of this proposal I will explore the consequences for rational Cherednik algebras at t=1 of a recent generalization by Kashiwara and Rouquier of the localization method. I aim to use the localization method to introduce powerful geometric techniques, such as the theory of perverse sheaves, to the study of the representation theory of rational Cherednik algebras. The skills developed here are applicable to many other objects currently of interest to representation theorists such as finite W-algebras and quantum Hamiltonian reduction of quiver varieties.