Hilbert Series for Moduli Spaces of Instantons on C²/Z_n

We study chiral gauge-invariant operators on moduli spaces of G instantons for any classical group G on A-type ALE spaces using Hilbert Series (HS). Moduli spaces of instantons on an ALE space can be realized as Higgs branches of certain quiver gauge theories which appear as world-volume theories on Dp branes in a Dp-D(p+4) system with the D(p+4) branes (with or without O(p+4) planes) wrapping the ALE space. We study in detail a list of quiver gauge theories which are related to G-instantons of arbitrary ranks and instanton numbers on a generic A_{n-1} ALE space and discuss the corresponding brane configurations. For a large class of theories, we explicitly compute the Higgs branch HS which reveals various algebraic/geometric aspects of the moduli space such as the dimension of the space, generators of the moduli space and relations connecting them. In a large number of examples involving lower rank instantons, we demonstrate that HS for equivalent instantons of isomorphic gauge groups but very different quiver descriptions do indeed agree, as expected.