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Research interests
I work in representation theory, with geometrical methods. I am particularly interested in the following topics:- Intersection cohomology complexes and perverse sheaves over the integers and in positive characteristic
- Lie theory (Weyl groups, reductive groups, root systems, Schubert calculus, nilpotent orbits)
- Springer correspondence
- Character sheaves
- Representations of reductive groups
- Affine Grassmannian, geometric Satake isomorphism
Modular Springer correspondence and decomposition matrices
Springer correspondence makes a link between the geometry of the nilpotent orbits in a Lie algebra and the (ordinary) irreducible representations of its Weyl group. Lusztig, Bohro and McPherson explained this in terms of intersection cohomology complexes on the variety of nilpotent elements.I defined a modular Springer correspondence and showed that the decomposition matrix of the Weyl group is a part of the decomposition matrix for equivariant perverse sheaves on the nilpotent variety. I determined the modular Springer correspondence in the case of the symmetric group, and in other types in rank up to three.
On the other hand, I calculated geometrically some decomposition numbers for perverse sheaves.
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I determined all the decomposition
numbers involving the regular and subregular classes,
using results of Brieskorn and Slodowy about the singularity of the
nilpotent variety along the subregular class.
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To study the modular
reduction of the intersection cohomology complex with constant
coefficients of the closure of the minimal orbit, I computed the
cohomology of the minimal nilpotent orbit
with integral coefficients. This calculation involved a Gysin sequence
for a line bundle over a generalised flag variety, Schubert calculus,
and the combinatorics of the root system.
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I showed that James' row and column removal rule for the decomposition
numbers of the symmetric groups is a consequence of
a smooth equivalence of singularities found by Kraft and Procesi.
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Kraft and Procesi also proved a row and removal rule for other
classical types, which gives immediatly partial information about
decomposition numbers for adjacent orbits (in the order given by
inclusion of closures). A further study should yield all the
decomposition numbers of this kind.
- They also have a result about the special decomposition of the nilpotent variety (as defined by Lusztig and Spaltenstein) that we can use. They proved that each special piece in the quotient of a smooth variety by an elementary abelian 2-group. We deduce that the modular reduction of the intersection cohomology complex with constant coefficients of the closure of a special class does not involve any simple perverse sheaf extending a local system on a smaller class in the same special piece, for odd primes.
So the problem of the determination of the decomposition matrices of the Weyl groups (including the symmetric groups) has been reduced to a geometrical problem about perverse sheaves on the nilpotent variety with coefficients in positive characteristic; actually, it would be enough to compute stalks of intersection cohomology complexes. We have been able to find geometrically some decomposition numbers. Perhaps one will be able to compute all decomposition numbers in this geometric setting, but of course this should be very difficult.
Lusztig generalized Springer correspondence and defined character sheaves to study ordinary irreducible representations of finite reductive groups. In the future, I would like to make a theory of modular character sheaves to study the modular representation theory of finite reductive groups.
Affine Grassmannian
I also studied decomposition numbers for perverse sheaves on the affine Grassmannian. Using the geometric Satake isomorphism of Mirkovic and Vilonen and the description of the minimal degenerations in the affine Grassmannian by Malkin, Ostrik and Vybornov, one can recover geometrically some decomposition numbers for reductive groups. On the other hand, one can also go in the other direction: I was able to prove that some singularities are not smoothly equivalent, using decomposition numbers for reductive groups. More importantly, I realized that my previous calculations on minimal singularities provide counter-examples to a conjecture of Mirkovic and Vilonen about the absence of torsion in IC stalks over Z in the affine Grassmannian. One can still make such a conjecture over Z_p for p good. A preprint is available here : Modular representations of reductive groups and geometry of affine Grassmannians.Parity sheaves
This is joint work with Carl Mautner and Geordie Williamson. See our preprint: Parity sheaves.Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case.
Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig's conjecture.
Kumar's criterion modulo p
Kumar's criterion (and its generalizations by Arabia and Brion) allows one to find the smooth and rationally smooth loci of a class of varities with a torus action, including Schubert varieties, using equivariant multiplicities. In Kumar's criterion modulo p, Geordie Williamson and I generalize this result to determine the p-smooth locus of such varieties.
