Timothy Logvinenko (Cardiff University)

Abstract: Two-dimensional McKay correspondence originated in the observation by John McKay that the representation graph of a finite subgroup G of SL_2(C) coincides with the Coxeter graph of an affine Lie algebra \mathfrak{g} of ADE type. It turned out that the combinatorics of \mathfrak{g} control not only the representation theory of G but also the geometry of the minimal resolution Y of C^2/G.

In the first half of the talk I will give a gentle introduction to the subject, illustrated by examples. We will review the finite subgroups of SL_2(C), the McKay quiver Q of G, the geometry of the minimal resolution Y, and its construction as a moduli space of semistable representations of Q. The stability parameter space \Theta with the stratification by the semistable walls coincides with the Cartan algebra \mathfrak{h} of \mathfrak{g} stratified by root hyperplanes. I will show how the reflections in the classes of the exceptional curves on Y define an action of the braid group B_{\mathfrak{g}} on the cohomology, K-theory, and the derived category D(Y) of Y.

In the second half of the talk, I will report on the ongoing project to construct a certain categorical structure on an affine hyperplane arrangement on \mathfrak{h} refining that of the root hyperplanes. The braid group action above can be viewed as a categorical local system with the fibre D(Y) on the open stratum of \mathfrak{h}/W, where W is the Weyl group. We aim to extend this to a W-equivariant categorical perverse sheaf, a “perverse schober”, on the whole of the affine hyperplane arrangement. This is joint work with Arman Sarikyan (LIMS).