Orbit closures in the enhanced nilpotent cone

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.Comment: 32 pages. Update (August 2010): There is an error in the proof of Theorem 4.7, in this version and the almost-identical published version. See the corrigendum arXiv:1008.1117 for independent proofs of later results that depend on that statemen

Mirabolic Robinson–Shensted–Knuth correspondence

“…From this we obtain that H m (IC(S)| Q ) = 0 for m > −n − ℓ(w ′ ) and dim H −n−ℓ(w ′ ) (IC(S)| Q ) = dim H −n−ℓ(w ′ )−2 (IC(S) x ′ ) where x ′ ∈ U . Note that if we identify U with φ(U ), we have IC(S)| U = IC(Ωw )| φ(U) [1]. Besides φ(U ) ⊂ Ωw′ * s .…”

“…I am grateful to P. Achar and A. Henderson for sending me [1] prior to its publication, where our Theorem 1 is proved independently (as Proposition 2.3). I am indebted to M. Finkelberg for posing the problem, numerous valuable discussions and help in editing the paper, and to G. Lusztig for pointing out the reference [18] which also classifies the orbits in Fl(V ) × Fl(V ) × V and studies R as an algebra.…”

“…The starting point of our work is a classification of GL(V )-orbits in N × V where N is the nilpotent cone in End(V ) (it was independently obtained by P. Achar and A. Henderson in [1]). We prove (see section 2.2) that the set of orbits is in a natural bijection with the set P of pairs of partitions (ν, θ) such that |ν| = N , and ν ⊃ θ, that is ν i ≥ θ i ≥ ν i+1 for any i ≥ 1.…”

Point stabilisers for the enhanced and exotic nilpotent cones

“…For example, it was shown by Achar-Henderson [1] that there is a bijection GnðV Â NÞ $ Q n and that the closure orderings on these orbits agree with those on KnðW Â N 0 Þ and are both given by a natural partial order on Q n . These similarities are extended by a conjecture of Achar-Henderson [1, Section 6], which claims that the local intersection cohomology for the exotic nilpotent cone is the same as that of the enhanced nilpotent cone but with twice the degree.…”

“…A recent innovation in this direction is the introduction of the GLðV Þ-set V Â N, which has appeared in the work of Achar-Henderson [1], Travkin [7] and others, and was given the name enhanced nilpotent cone by Achar-Henderson in [1]. The enhanced cone mimics the cominatorics of the exotic cone while being a more accessible object.…”