Simon Riche

Koszul duality and modular representations of semisimple Lie algebras

In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen and Soergel). We also give information about the Koszul dual rings. Our main tool is the localization theory in positive characteristic developed by Bezrukavnikov, Mirkovic and Rumynin.

Modular Koszul duality

We prove an analogue of Koszul duality for category O of a reductive group G in positive characteristic p larger than 1 plus the number of roots of G. However there are no Koszul rings, and we do not prove an analogue of the Kazhdan--Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of G plus 2.

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.

Linear Koszul duality, II: coherent sheaves on perfect sheaves

In this paper we continue the study (initiated in a previous article of the authors) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.

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

Localization of modules for a semisimple Lie algebra in prime characteristic (with an Appendix by R. Bezrukavnikov and S. Riche: Computation for sl(3))

G_2

Tilting modules and the p-canonical basis

G2.

Linear Koszul duality

We study some aspects of modular generalized Springer theory for a complex reductive group G with coefficients in a field