Inversion formulae in motivic Hall algebras

Given a variety $X$, the Hilbert scheme of points on $X$ is a much-studied example of a moduli space. A way to understand this Hilbert scheme better, is to compute its "baby motive", namely its class in the Grothendieck ring of varieties. In this talk, we will present a strategy to compute these classes when $X$ is a Kleinian singularity, using motivic Hall algebras.

In our setting, such an algebra is defined using a quiver $Q$. We can consider the stack $M$ of all finite-dimensional $Q$-representations, and consider the Grothendieck ring of stacks over $M$. In this ring, we can define a Hall product by "convolution along extensions", and this gives us the motivic Hall algebra of $Q$-representations. Choosing a stability condition for $Q$-representations, we can then consider the class of the stack of semistable representations (of some fixed slope). In a completion of the motivic Hall algebra, such a class is invertible, and we present a formula for its inverse. We then explain how to conjecturally apply this to our original problem of computing classes in the Grothendieck ring of varieties.
This is work in progress, joint with Okke van Garderen.