A reducible characteristic variety in type A

We show that simple highest weight modules for sl_12 may have reducible characteristic variety. This answers a question of Borho-Brylinski and Joseph from 1984. The relevant singularity under Beilinson-Bernstein localization is the (in)famous Kashiwara-Saito singularity. We sketch the rather indirect route via the p-canonical basis, W-graphs and decomposition numbers for perverse sheaves that led us to examine this singularity.Comment: 13 pages, comments welcom

On an analogue of the James conjecture

We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs in type A_5 for p = 2 and involves the same singularity used by Kashiwara and Saito to show the reducibility of the characteristic variety of an intersection cohomology D-module on a quiver variety. Using recent results of Polo one can give counterexamples in type A in all characteristics.Comment: 12 pages. v2: final versio

Local Hodge theory of Soergel bimodules

We prove the local hard Lefschetz theorem and local Hodge-Riemann bilinear relations for Soergel bimodules. Using results of Soergel and K\"ubel one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.Comment: 54 pages, v3: further minor changes. Final version to appear in Acta. Mat

The anti-spherical category

We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localisation procedure for the anti-spherical category, from which we construct a "light leaves" basis of morphisms. Our techniques may be used to calculate many new elements of the $p$-canonical basis in the anti-spherical module.Comment: Best viewed in colo