We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C 2 /G], respectively the singular quotient surface C 2 /G, where G < SL(2, C) is a finite subgroup of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.

Abstract. 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 < SL(2, C) a finite subgroup, 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. Euler characteristics of Hilbert schemes of pointsLet X be a quasiprojective variety X over the field C of complex numbers. Let Hilb m (X) denote the Hilbert scheme of m points on X, the quasiprojective scheme parametrizing 0-dimensional subschemes of X of length m. Consider the generating series of topological Euler characteristicsFor a smooth variety X, the series Z X (q), as well as various refinements, have been extensively studied. For a nonsingular curve X = C, we have MacDonald's result [8]For a nonsingular surface X = S, we have (a specialization of) Göttsche's formula [5] (1).There are also results for higher-dimensional varieties [1]. For singular varieties X, very little is known about the series Z X (q). For a singular curve X = C with a finite set {P 1 , . . . , P k } of planar singularities however, we have the beautiful conjecture of Oblomkov and Shende [14], proved by Maulik [9], which specializes to the following:Here each Z (Pi,C) (q) is a highly nontrivial local term that can be expressed in terms of the HOMFLY polynomial of the embedded link of the singularity P i ∈ C. Simple surface singularitiesIn this announcement, we consider the generating series Z S (q) for X = S a singular surface with simple (Kleinian, rational double point) singularities. First, we discuss the local situation. As is well known, locally analytically S is a quotient singularity S = C 2 /G ∆ . Here G ∆ < SL(2, C) is a finite subgroup corresponding to an irreducible simply-laced Dynkin diagram ∆, the dual graph of the exceptional components in the minimal resolution of the singularity. There are three possible types: ∆ can be of type A n for n ≥ 1, type D n for n ≥ 4 and type E n for n = 6, 7, 8. The following is our main conjecture.

For a finite subgroup Γ ⊂ SL(2, C) and n 1, we construct the (reduced scheme underlying the) Hilbert scheme of n points on the Kleinian singularity C 2 /Γ as a Nakajima quiver variety for the framed McKay quiver of Γ, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by removing an arrow and then 'cornering', and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of the stability parameter.