  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.
