It is well known that GLn(C)-orbits in the
nilpotent cone N (consisting of n×n
nilpotent complex matrices) are parametrized by partitions
of n. Some of the geometry of the orbit closures is
reflected in their intersection cohomology polynomials,
which were shown by Lusztig to equal the combinatorial Kostka-Foulkes polynomials.
The `enhanced nilpotent cone' of the title is nothing but the product
(consisting of pairs of a vector and a nilpotent matrix).
The obvious action of GLn(C) still has
finitely many orbits, now parametrized by bipartitions of n.
I will discuss the closures of these orbits, and a conjecture
on their intersection cohomology polynomials.
This is joint work with Pramod Achar (Louisiana State University).