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

Abstract. We define a generalized Springer correspondence for the group GL(n) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL(n) satisfying the 'recollement' properties, and with subquotients equivalent to categories of representations of a product of symmetric groups.

Abstract. We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-0 coefficients. We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for SL(n) with coefficients of arbitrary characteristic and for SO(n) and Sp(2n) with characteristic-2 coefficients.

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the `split-quotient quiver' introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras.Comment: 43 pages. In version 2, at the referee's suggestion, we slightly expand some statements (Theorem 1.2 and Proposition 3.19) to include the relevant affine varieties. This version is to appear in Advances in Mathematic

Abstract. For a simply-connected simple algebraic group G over C, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of G, generalizing a well-known fact about GLn. Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.