Smooth group quotients and the Chevalley--Shephard--Todd theorem

When Z/2Z acts on C^2 by reflection in an axis, the quotient C^2 / (Z/2Z) is smooth and isomorphic to C^2. When the action is by rotation by pi, the quotient is not smooth--it is a cone with a singularity at the origin. The general picture is provided by the Chevalley—Shephard—Todd theorem, which states that (e.g.) a finite group G acting linearly on C^n gives rise to a smooth quotient if and only if G is generated by pseudoreflections: elements with fixed-point set of codimension 1. The Chevalley—Shephard—Todd theorem provides a smoothness characterization in the land of Deligne—Mumford stacks. It is natural to ask: what happens in the more general case of good moduli spaces of Artin stacks? In this talk I will present a conjectural generalization of the Chevalley—Shephard—Todd theorem to the reductive group setting, and give the idea of a proof in the case of irreducible representations of simple Lie groups via the étale slice theorem and some counting and convexity arguments. This is joint work with Dan Edidin and Matthew Satriano. No algebraic geometry background will be assumed (but you may want to remind yourself what a group is).