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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.
Schubert varieties
I recently 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, I was able to recover geometrically some decomposition numbers for reductive groups. This uses previous work on the nilpotent variety. On the other hand, I 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. A preprint is available here : Modular representations of reductive groups and geometry of affine Grassmannians.Besides, I have an ongoing project with Geordie Williamson about the relationship between the integer appearing as the numerator of an equivariant multiplicity in Kumar's criterion for rational smoothness on Schubert varieties, and the torsion in the intersection cohomology stalks (both in the finite and affine cases).
