  This is an announcement of conjectures and results concerning the generating
series of Euler characteristics of Hilbert schemes of points on surfaces with
simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite
subgroup of SL(2, C), we conjecture a formula for this generating series in
terms of Lie-theoretic data, which is compatible with existing results for type
A singularities. We announce a proof of our conjecture for singularities of
type D. The generating series in our conjecture can be seen as a specialized
character of the basic representation of the corresponding (extended) affine
Lie algebra; we discuss possible representation-theoretic consequences of this
fact. Our results, respectively conjectures, imply the modularity of the
generating function for surfaces with type A and type D, respectively
arbitrary, simple singularities, confirming predictions of S-duality.
