Logarithmic differential forms on Bott-Samelson varieties and braid relations

Braid relations in the coherent version of the affine Hecke category were established via a delicate study of Steinberg variety, by Bezrukavnikov and Riche. The talk is devoted to an alternative approach to establishing braid relations in a version of affine Hecke category developed in the thesis of my student Sebastian Orsted.

We begin with proposing a realization of affine Hecke category Koszul dual to the one of Bezrukavnikov-Riche: we define a category of equivariant modules over the ring of differential forms on a reductive algebraic group G, equipped with convolution monoidal structure. Then we introduce the candidates for the braid group generators given by DG-modules of logarithmic differential forms on the minimal parabolic subgroups. The proof of braid relations goes via the study of logarithmic differential forms on large Bott-Samelson varieties and is based on the following observation.

Let X be a desingularization of the variety Y such that the preimage of a divisor containing singularities of Y in X is a divisor with normal crossings. Then the direct image of the sheaf of logarithmic differential forms on X to Y depends on Y, not on the resolution of singularities X.

We outline the calculation leading to the proof of this statement.