The Springer correspondence in type A provides a bijection between nilpotent orbits for the general linear group and complex, finite-dimensional, irreducible representations (up to isomorphism) of the symmetric group. Outside of type A the Springer correspondence is more complicated and involves additional data arising from the component group. In his work on the representation theory of multiparameter Hecke algebras of type C, Kato obtains a bijection between orbits in the exotic nilpotent cone and the complex, finite-dimensional, irreducible representations of the Weyl group of type C which does not involve the component group. This bijection is obtained by constructing an action of the Weyl group on the cohomology of so-called exotic Springer fibers.
In this talk we will explain this in more detail and study the geometry and topology of a certain family of exotic Springer fibers and describe our results in terms of the combinatorics which appears in the study of the two-boundary Temperley-Lieb algebra. If time permits we will discuss how this might be related to interesting categorification problems arising in low-dimensional topology. This is joint work with V. Nandakumar and N. Saunders.