Article

# Representation theory of rectangular finite $W$-algebras

03/2010; DOI: 10.1016/j.jalgebra.2011.05.014

Source: arXiv

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**ABSTRACT:**We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated P^1-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type D diagram calculus labelling the irreducible components in an convenient way which relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types. The results will be connected to Brauer algebras at non-generic parameters in a subsequent paper.09/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**A highest weight theory for a finite W-algebra U(g,e) was developed in [BGK]. This leads to a strategy for classifying the irreducible finite dimensional U(g,e)-modules. The highest weight theory depends on the choice of a parabolic subalgebra of g leading to different parameterizations of the finite dimensional irreducible U(g,e)-modules. We explain how to construct an isomorphism preserving bijection between the parameterizing sets for different choices of parabolic subalgebra when g is of type A, or when g is of types C or D and e is an even multiplicity nilpotent elementJournal of Algebraic Combinatorics 05/2011; · 0.63 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We complete the classification of the finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent elements in classical Lie algebras. This extends earlier work where this classification is determined for such representations of integral central character.12/2011;

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