Geordie Williamson

A geometric model for Hochschild homology of Soergel bimodules

An important step in the calculation of the triply graded link homology of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL.n/. We present a geometric model for this Hochschild homology for any simple group G , as B ‐equivariant intersection cohomology of B B ‐orbit closures in G . We show that, in type A, these orbit closures are equivariantly formal for the conjugation B ‐action. We use this fact to show that, in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and to describe its Hilbert series, proving a conjecture of Jacob Rasmussen. 17B10; 57T10

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.

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.

Diagrammatics for Coxeter groups and their braid groups

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.

On the character of certain tilting modules

Let G be a semisimple group over an algebraically closed field of characteristic p > 0. We give a (partly conjectural) closed formula for the character of many indecomposable tilting rational G-modules assuming that p is large.

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.

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 Kübel, 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.

Standard objects in 2-braid groups

For any Coxeter system we establish the existence (conjectured by Rouquier) of analogues of standard and costandard objects in 2-braid groups. This generalizes a known extension vanishing formula in the BGG category O.

Kumar’s criterion modulo

We prove that equivariant multiplicities may be used to determine whether attractive fixed points on T-varieties are p-smooth. This gives a combinatorial criterion for the determination of the p-smooth locus of Schubert varieties for all primes p.

p

In this note we explain how Lusztig’s induction and restriction functors yield categorical actions of Kac-Moody algebras on the derived category of unipotent representations. We focus on the example of finite linear groups and induction/restriction associated with split Levi subgroups, providing a derived analogue of Harish-Chandra induction/restriction as studied by Chuang-Rouquier in [5]. 2010 Mathematics Subject Classification. Primary 20C33.We study composition series of derived module categories in the sense of Angeleri Hugel, Konig & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confirms part of a conjecture of Bobinski & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of different lengths and composition factors. In other words, there is no Jordan-Holder property for composition series of derived categories of quasi-hereditary algebras.We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are