Representation theory of finite W-algebras

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Recently there has been a great deal of interest in finite W-algebras and their representation theory from world leading mathematicians. This is largely due to connections to the representation theory and geometry of finite dimensional reductive Lie algebras. Finite W-algebras and their affine counterparts have also attracted much interest in mathematical physics motivated by their occurrence as non-linear symmetry algebras in conformal field theory. The purpose of this research is to make significant advances in the representation theory of finite W-algebras and in particular to resolve the important problem of classifying finite dimensional simple modules.In recent joint work Brundan, Kleshchev and the author, have developed a highest weight theory for finite W-algebras. This leads to a natural strategy for tackling the problem of classifying finite dimensional simple modules, in analogy with the situation for complex reductive Lie algebras. The primary aim of this proposal is to successfully follow this strategy. The recent developments in the area of finite W-algebras and the diverse areas of mathematics feeding in to their representation theory will allow us to employ a variety of techniques in achieving this goal. In particular, we intend to exploit the interplay between the three equivalent definitions of a finite W-algebra: the Whittaker model definition, the definition via quantum Hamiltonian reduction, and the definition via Fedosov reduction. In addition to our main objective, there are a number of other important problems concerning the representation theory of finite W-algebras that we will investigate.