Koszul duality and Frobenius structure for restricted enveloping algebras

Let g be the Lie algebra of a connected, simply connected semisimple algebraic group over an algebraically closed field of sufficiently large positive characteristic. We study the compatibility between the Koszul grading on the restricted enveloping algebra (Ug)_0 of g constructed in a previous paper, and the structure of Frobenius algebra of (Ug)_0. This answers a question raised to the author by W. Soergel.Comment: 30 page

Tilting modules and the p-canonical basis

In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke category of a general crystallographic Coxeter group may be described in terms of parity complexes on the flag variety of the corresponding Kac-Moody group.Comment: 145 pages. Many TikZ figures (best viewed in colour). v3: many minor changes, detail and references for Kac-Moody flag varieties adde

Iwahori-Matsumoto involution and linear Koszul Duality

We use linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras studied in previous papers of the authors to give a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras. More generally we prove that linear Koszul duality is compatible with convolution in a general context related to convolution algebras.Comment: v1: 29 pages, the present paper supersedes arXiv:0903.0678; v2: 26 pages, minor modifications; v3: 29 pages, final version, published in IMR

Linear Koszul duality and Fourier transform for convolution algebras

In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in K-homology defined in a previous paper and the Fourier transform isomorphism for convolution algebras in Borel-Moore homology are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the categorification of the Iwahori-Matsumoto involution for graded affine Hecke algebras (due to Evens and the first author) and for usual affine Hecke algebras (obtained in a previous paper).Comment: v1: 29 pages; v2: 41 pages, many details added; v3: 42 pages, minor modifications (final version, to appear in Doc. Math.

Modular Koszul duality

We prove an analogue of Koszul duality for category $\mathcal{O}$ of a reductive group $G$ in positive characteristic $\ell$ 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.Comment: 62 pages; image displays best in pd