Orbit closures in the enhanced nilpotent cone

TL;DR

This paper investigates the orbit structure and closure relations in the enhanced nilpotent cone, revealing connections to bipartition combinatorics, intersection cohomology, and Kato's exotic nilpotent cone, advancing understanding of orbit closures in representation theory.

Contribution

It establishes a correspondence between orbit closures and bipartition partial orders, and links intersection cohomology to bipartition analogues of Kostka polynomials, also relating to Kato's exotic cone.

Findings

Closure ordering matches bipartition partial order.

Intersection cohomology relates to bipartition Kostka polynomials.

Connection with Kato's exotic nilpotent cone confirmed.

Abstract

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.