Orbit closures in the enhanced nilpotent cone

Details Orbit closures in the enhanced nilpotent cone Orbit closures in the enhanced nilpotent cone

2007-12-07

arxiv.org/abs/0712.1079v2

Adv. Math. 219 (2008), no. 1, pp. 27-62

Mathematics

Representation Theory

32 pages. Update (August 2010): There is an error in the proof of Theorem 4.7, in this version and the almost-identical publis

Scientific paper

10.1016/j.aim.2008.04.008

We study the orbits of $G=\mathrm{GL}(V)$ in the enhanced nilpotent cone $V\times\mathcal{N}$, where $\mathcal{N}$ is the variety of nilpotent endomorphisms of $V$. These orbits are parametrized by bipartitions of $n=\dim V$, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled.

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