Perverse schobers and the McKay correspondence

I will talk about constructing a perverse schober, a poor man’s perverse sheaf of triangulated categories, in the context of the classical McKay correspondence for $G \subset SL_2(C)$. The braid group of the corresponding ADE type acts on the derived category D(Y) of the minimal resolution Y of C^{^2}/G by spherical twists in the exceptional curves. This braid group is the fundamental group of the open stratum of $\mathfrak{h}/W$, the quotient of the ADE Cartan algebra by the Weil group action, so its action on D(Y) can be thought of as a local system of triangulated categories on this open stratum. A perverse schober extends this structure to the higher codimension stratas. We actually construct a W-equivariant schober on $\mathfrak{h}$ by using an instance of the McKay correspondence – the root hyperplane arrangement in $\mathfrak{h}$ coincides with the wall-and-chamber structure in the GIT stability space for the construction of Y as the moduli space of the McKay quiver representations. The schober we construct on the GIT stability space neatly packages up all the GIT wall-crossing equivalences and more. Our work is motivated by wanting to eventually tackle dim=3 case, where $h/W$ picture no longer exists, however, it might still be possible to construct a similar schober on the GIT stability space. This is a joint work with Arman Sarikyan (LIMS).