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.

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 statement

References in corpus (3)

• Mirabolic affine Grassmannian and character sheaves
• Cherednik algebras for algebraic curves
• An exotic Springer correspondence for symplectic groups

Cited by in corpus (31)

• An exotic Deligne-Langlands correspondence for symplectic groups
• Mirabolic affine Grassmannian and character sheaves
• An algebraic study of extension algebras
• The exotic Robinson-Schensted correspondence
• Cherednik algebras for algebraic curves
• Moduli spaces, indecomposable objects and potentials over a finite field
• Corrigendum to `Orbit closures in the enhanced nilpotent cone', published in Adv. Math. 219 (2008)
• Orthosymplectic Satake equivalence
• Classic and mirabolic Robinson-Schensted-Knuth correspondence for partial flags
• An exotic Springer correspondence for symplectic groups
• Quiver Hall-Littlewood functions and Kostka-Shoji polynomials
• Equivariant coherent sheaves on the exotic nilpotent cone
• Irreducible components of exotic Springer fibres
• The mirabolic Hecke algebra
• Point stabilisers for the enhanced and exotic nilpotent cones
• Normality of orbit closures in the enhanced nilpotent cone
• On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras
• Springer correspondence for symmetric spaces
• Irreducible Components of Exotic Springer fibres II: The Exotic Robinson-Schensted Algorithm
• Pieces of nilpotent cones for classical groups
• Enhanced Nilpotent Representations of a Cyclic Quiver
• Enhanced adjoint action and their orbits for the general linear group
• Semi-simplicity of the category of admissible D-modules
• On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures
• $q$-Kreweras numbers for coincidental Coxeter groups attached to limit symbols
• Exotic Springer fibers for orbits corresponding to one-row bipartitions
• Springer correspndence for complex reflection groups
• Exotic symmetric spaces of higher level - Springer correspondence for complex reflection groups -
• Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov
• A class of perverse sheaves on framed representation varieties of the Jordan quiver
• Two-row Delta Springer varieties