Mass formulas for local Galois representations and quotient singularities II: dualities and resolution of singularities

A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper, the authors observed that in a particular example, two total masses coming from two different weightings are dual to each other. We discuss the problem how general such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincar\'e duality for stringy invariants. We also exhibit several examples.