Anthony Henderson

Orbit closures in the enhanced nilpotent cone

We study the orbits of G=GL(V) in the enhanced nilpotent cone , where is the variety of nilpotent endomorphisms of V. These orbits are parametrized by bipartitions of n=dimV, 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 Katos 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

Weyl group actions on the Springer sheaf

We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence.

Representations of wreath products on cohomology of De Concini-Procesi compactifications

The wreath product W(r,n) of the cyclic group of order r and the symmetric group S n acts on the corresponding projective hyperplane complement and on its wonderful compactification as defined by De Concini and Procesi. We give a formula for the characters of the representations of W(r,n) on the cohomology groups of this compactification, extending the result of Ginzburg and Kapranov in the r=1 case. As a corollary, we get a formula for the Betti numbers, which generalizes the result of Yuzvinsky in the r=2 case. Our method involves applying to the nested-set stratification a generalization of Joyals theory of tensor species, which includes a link between polynomial functors and plethysm for general r. We also give a new proof of Lehrers formula for the representations of W(r,n) on the cohomology groups of the hyperplane complement.

Fourier transform, parabolic induction, and nilpotent orbits

We prove that in the symmetric space setting the functors of Fourier transform (in the sense of Deligne) and parabolic induction (in the sense of Lusztig) commute. We derive two consequences: the first is a new proof of Lusztigs description of the intersection cohomology of nilpotent orbit closures for GLn, and the second is an analogous description for GL2n/Sp2n.

Diagram automorphisms of quiver varieties

Abstract 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.

Rational cohomology of the real Coxeter toric variety of type A

The toric variety corresponding to the Coxeter fan of type A can also be described as a De Concini-Procesi wonderful model. Using a general result of Rains which relates cohomology of real De Concini-Procesi models to poset homology, we give formulas for the Betti numbers of the real toric variety, and the symmetric group representations on the rational cohomologies. We also show that the rational cohomology ring is not generated in degree 1.

The exotic Robinson–Schensted correspondence☆

Abstract We study the action of the symplectic group on pairs of a vector and a flag. Considering the irreducible components of the conormal variety, we obtain an exotic analogue of the Robinson–Schensted correspondence. Conjecturally, the resulting cells are related to exotic character sheaves.

The Cohomology of Real De Concini–Procesi Models of Coxeter Type

We study the rational cohomology groups of the real De Concini-Procesi model corresponding to a finite Coxeter group, generalizing the type-A case of the moduli space of stable genus 0 curves with marked points. We compute the Betti numbers in the exceptional types, and give formulae for them in types B and D. We give a generating-function formula for the characters of the representations of a Coxeter group of type B on the rational cohomology groups of the corresponding real De Concini-Procesi model, and deduce the multiplicities of one-dimensional characters in the representations, and a formula for the Euler character. We also give a moduli space interpretation of this type-B variety, and hence show that the action of the Coxeter group extends to a slightly larger group.

MODULAR GENERALIZED SPRINGER CORRESPONDENCE III: EXCEPTIONAL GROUPS

We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type