$G$ fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products
Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let Z_{X,G}(q) \sum_{n=0}\infty} e(\operatorname{Hilb}{n}(X){G} q^{n 1} be the generating function for the Euler characteristics of Hilbert scheme of $G$ invariant length $n$ subschemes. We show that its reciprocal, $Z_{X,G}(q) 1} is the Fourier expansion of a modular cusp form of weight \frac{1}{2} e(X/G) for the congruence subgroup \Gamma_{0}|G| We give an explicit formula for $Z_{X,G} in terms of the Dedekind eta function for all 82 possible (X,G) The key intermediate result we prove is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system.