MSI Weekly Bulletin - Week starting Monday 5 November, 2007

[Joint work with Pramod Achar (Louisiana State University)] The ordinary nilpotent cone $\mathcal{N}_n$ consists of $n\times n$ nilpotent complex matrices. By the Jordan Canonical Form theorem, the $GL_n(\mathbb{C})$-orbits in this variety (i.e. similarity classes of nilpotent matrices) are parametrized by partitions of $n$ -- the same set that indexes the irreducible representations of the symmetric group $S_n$. This is more than just a coincidence, because the representations of $S_n$ can be constructed in terms of the topology of the orbit closures, as was shown by Springer and Lusztig. In this talk I will consider the `enhanced nilpotent cone' $\mathbb{C}^n\times\mathcal{N}_n$, consisting of pairs of a vector and a nilpotent matrix. I will explain why the $GL_n(\mathbb{C})$-orbits are parametrized by bipartitions of $n$, give a combinatorial description of the closures of these orbits, and discuss conjectural relations to Kato's `exotic nilpotent cone' and to the representations of the Coxeter group of type $B_n$.

TBA