  Working from first principles, quantization of a class of Hamiltonian systems
with reducible symmetry is carried out by constructing first the appropriate
reduced phase space and then the BRST cohomology. The constraints of this
system correspond to a first class set for a group G and a second class set for
a subgroup H. The BRST operator constructed is equivariant with respect to H.
Using algebraic techniques analogous to those of equivariant de Rham theory,
the BRST operator is shown to correspond to that obtained by BV quantization of
a class of systems with reducible symmetry. The 'ghosts for ghosts' correspond
to the even degree two generators in the Cartan model of equivariant
cohomology. As an example of the methods developed, a topological model is
described whose BRST quantization relates to the equivariant cohomology of a
manifold under a circle action.
