Hilbert series for moduli spaces of two instantons

The Hilbert Series (HS) of the moduli space of two G instantons on C ², where G is a simple gauge group, is studied in detail. For a given G, the moduli space is a singular hyperKähler cone with a symmetry group U(2) × G, where U(2) is the natural symmetry group of C². Holomorphic functions on the moduli space transform in irreducible representations of the symmetry group and hence the Hilbert series admits a character expansion. For cases that G is a classical group (of type A, B, C, or D), there is an ADHM construction which allows us to compute the HS explicitly using a contour integral. For cases that G is of E-type, recent index results allow for an explicit computation of the HS. The character expansion can be expressed as an infinite sum which lives on a Cartesian lattice that is generated by a small number of representations. This structure persists for all G and allows for an explicit expressions of the HS to all simple groups. For cases that G is of type G ₂ or F ₄, discrete symmetries are enough to evaluate the HS exactly, even though neither ADHM construction nor index is known for these cases.