Constructible sheaves on nilpotent cones in rather good characteristic

We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $mathbb k$ under the assumption that the characteristic $ell$ of $mathbb k$ is rather good for $G$, i.e., $ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautners characteristic-$ell$ `cleanness conjecture; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.

Parabolic degeneration of rational Cherednik algebras

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between two standard modules. The latter condition reproduces and enhances, in the case of the symmetric group, the combinatorics of cores and dominance order, and in general shows that the c-ordering on category O may be replaced by a much coarser ordering. The former gives a new proof of the classification of finite dimensional irreducible modules for the Cherednik algebra of the symmetric group.

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 $A_{2k-1}$. 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 $SL_2(mathbb C)$-variety whose normalization is ${mathbb A}^2$, an $Sp_4(mathbb C)$-variety whose normalization is ${mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. 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 Slodowys work for the regular nilpotent orbit.

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 (defined over a base field of odd characteristic $p$, and with coefficients in a field of odd characteristic $ell eq p$): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We also provide a quicker proof, by sorting characters according to the dimension of the corresponding Springer fiber, an invariant which is directly computable from symbols.

Modular Springer correspondence, decomposition matrices and basic sets

The Springer correspondence makes a link between the characters of a Weyl group and the geometry of the nilpotent cone of the corresponding semisimple Lie algebra. In this article, we consider a modular version of the theory, and show that the decomposition numbers of a Weyl group are particular cases of decomposition numbers for equivariant perverse sheaves on the nilpotent cone. We give some decomposition numbers which can be obtained geometrically. In the case of the symmetric group, we show that James row and column removal rule for the symmetric group can be derived from a smooth equivalence between nilpotent singularities proved by Kraft and Procesi. We give the complete structure of the Springer and Grothendieck sheaves in the case of $SL_2$. Finally, we determine explicitly the modular Springer correspondence for exceptional types.

Modular generalized Springer correspondence II: classical groups

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 $mathrm{SL}(n)$ with coefficients of arbitrary characteristic and for $mathrm{SO}(n)$ and $mathrm{Sp}(2n)$ with characteristic-$2$ coefficients.