| Résume | The Springer correspondence for the general linear group is a perfect bijection between irreducible representations of the Weyl group $S_n$ and orbits in the nilpotent cone. For other types, such as type C (the symplectic group), it is less clean: nilpotent orbits must be replaced by pairs of an orbit and an equivariant simple local system, and not all of these pairs participate in the bijection. In 2006, Syu Kato revealed an exotic nilpotent cone of type C which leads to a bijection of the same form as the Springer correspondence in type A. The geometry of this exotic nilpotent cone has many remarkable features. The closure order on the orbits and (conjecturally) the intersection cohomology of the closures are determined by combinatorics which was introduced by Shoji in his study of `limit symbols'. The cone can be decomposed into smooth unions of orbits in two different ways, one corresponding to orbits in the usual type-B cone and the other corresponding to orbits in the usual type-C cone; in each case there is a coincidence of numerical invariants, which can be explained by considering what happens in characteristic 2. This is joint work with Pramod Achar and Eric Sommers. |
