51 research outputs found
Decomposition numbers for perverse sheaves
The purpose of this article is to set foundations for decomposition numbers
of perverse sheaves, to give some methods to calculate them in simple cases,
and to compute them concretely in two situations: for a simple (Kleinian)
surface singularity, and for the closure of the minimal non-trivial orbit in a
simple Lie algebra.
This work has applications to modular representation theory, for Weyl groups
using the nilpotent cone of the corresponding semisimple Lie algebra, and for
reductive algebraic group schemes using the affine Grassmannian of the
Langlands dual group
Springer basic sets and modular Springer correspondence for classical types
We define the notion of basic set data for finite groups (building on the
notion of basic set, but including an order on the irreducible characters as
part of the structure), and we prove that the Springer correspondence provides
basic set data for Weyl groups. Then we use this to determine explicitly the
modular Springer correspondence for classical types (for representations in odd
characteristic). In order to do so, we compare the order on bipartitions
introduced by Dipper and James with the order induced by the Springer
correspondence.Comment: 31 page
Modular representations of reductive groups and geometry of affine Grassmannians
By the geometric Satake isomorphism of Mirkovic and Vilonen, decomposition
numbers for reductive groups can be interpreted as decomposition numbers for
equivariant perverse sheaves on the complex affine Grassmannian of the
Langlands dual group. Using a description of the minimal degenerations of the
affine Grassmannian obtained by Malkin, Ostrik and Vybornov, we are able to
recover geometrically some decomposition numbers for reductive groups. In the
other direction, we can use some decomposition numbers for reductive groups to
prove geometric results, such as a new proof of non-smoothness results, and a
proof that some singularities are not equivalent (a conjecture of Malkin,
Ostrik and Vybornov). We also give counterexamples to a conjecture of Mirkovic
and Vilonen stating that the stalks of standard perverse sheaves over the
integers on the affine Grassmannian are torsion-free, and propose a modified
conjecture, excluding bad primes.Comment: 13 page
Generic singularities of nilpotent orbit closures
According to a well-known theorem of Brieskorn and Slodowy, the intersection
of the nilpotent cone of a simple Lie algebra with a transverse slice to the
subregular nilpotent orbit is a simple surface singularity. At the opposite
extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is
also an isolated symplectic singularity, called a minimal singularity. For
classical Lie algebras, Kraft and Procesi showed that these two types of
singularities suffice to describe all generic singularities of nilpotent orbit
closures: specifically, any such singularity is either a simple surface
singularity, a minimal singularity, or a union of two simple surface
singularities of type . In the present paper, we complete the picture
by determining the generic singularities of all nilpotent orbit closures in
exceptional Lie algebras (up to normalization in a few cases). We summarize the
results in some graphs at the end of the paper.
In most cases, we also obtain simple surface singularities or minimal
singularities, though often with more complicated branching than occurs in the
classical types. There are, however, six singularities which do not occur in
the classical types. Three of these are unibranch non-normal singularities: an
-variety whose normalization is , an
-variety whose normalization is , and a
two-dimensional variety whose normalization is the simple surface singularity
. In addition, there are three 4-dimensional isolated singularities each
appearing once. We also study an intrinsic symmetry action on the
singularities, in analogy with Slodowy's work for the regular nilpotent orbit.Comment: 56 pages (5 figures). Minor corrections. Accepted in Advances in Mat
Modular generalized Springer correspondence: an overview
This is an overview of our series of papers on the modular generalized
Springer correspondence. It is an expansion of a lecture given by the second
author in the Fifth Conference of the Tsinghua Sanya International Mathematics
Forum, Sanya, December 2014, as part of the Master Lecture `Algebraic Groups
and their Representations' Workshop honouring G. Lusztig. The material that has
not appeared in print before includes some discussion of the motivating idea of
modular character sheaves, and heuristic remarks about geometric functors of
parabolic induction and restriction.Comment: 19 pages. Version 2 includes more examples and tables in Section
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