Many features of an algebraic group are controlled
by the geometry of its nilpotent cone, which in the case of GLn(C)
is merely the variety N of n×n nilpotent matrices. The study of the
orbits of the group in its nilpotent cone leads to combinatorial data relating
to the representations of the Weyl group, via the famous Springer correspondence.
As I announced in an Algebra Seminar last year, Pramod Achar and I have shown that studying the orbits of GLn(C) in the enhanced nilpotent cone Cn×N leads to exotic combinatorial data of type B/C (previously defined by Shoji under the name of "limit symbols"). This is closely related to Syu Kato's exotic Springer correspondence for the symplectic group.
In the current Seminar, I will review this story and bring it up to date.