Equivariant Hilbert scheme of points on K3 surfaces and modular forms

Let X be a K3 surface and let Z_X(q) be the generating series of the topological Euler characteristics of the Hilbert scheme of points on X. It is known that q/Z_X(q) equals the discriminant form Delta(\tau) after the change of variables q=e^{2 \pi i \tau}. In this talk we consider the equivariant generalization of this results, when a finite group G acts on X symplectically. Mukai and Xiao have shown that there are exactly 81 possibilities for such an action in terms of types of the fixed points. The analogue of q/Z_X(q) in each 81 case turns out to be a modular form (after the same change of variables). Knowledge of modular forms is not assumed in the talk; I will introduce all necessary concepts. Joint work with Jim Bryan.