  We study the moduli spaces of self-dual instantons on CP^2 in a simple group
G. When G is a classical group, these instanton solutions can be realised using
ADHM-like constructions which can be naturally embedded into certain three
dimensional quiver gauge theories with 4 supercharges. The topological data for
such instanton bundles and their relations to the quiver gauge theories are
described. Based on such gauge theory constructions, we compute the Hilbert
series of the moduli spaces of instantons that correspond to various
configurations. The results turn out to be equal to the Hilbert series of their
counterparts on C^2 upon an appropriate mapping, in agreement with the result
of [arXiv:0802.3120]. We check the former against the Hilbert series derived
from the blowup formula for the Hirzebruch surface F_1 and find an agreement.
The connection between the moduli spaces of instantons on such two spaces is
explained in detail.
