No abstract

We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan–Lusztig polynomials and we prove it for finite simply-laced Weyl groups.

In [SZJ12] the parabolic Kazhdan-Lusztig polynomials for Grassmannians are computed by counting certain Dyck partitions. We "lift" this combinatorial formula to the intersection cohomology of Schubert varieties in Grassmannians and we obtain many bases of the intersection cohomology which extend (after dualizing) the classical Schubert basis of the ordinary cohomology.

Abstract. We introduce the Néron-Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the Néron-Severi Lie algebra to provide an easy proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality.

We generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules by taking minimal complexes of the restriction of Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier complexes: they are ∆-split, they satisfy a vanishing formula and, when Soergel's conjecture holds they are perverse. As an application, we use singular Rouquier complexes to establish Hodge theory for singular Soergel bimodules.