  There is an elaborated abstract form of BRST quantization on inner product
spaces within the operator formalism which leads to BRST invariant states of
the form |ph>=e^{[Q,\psi]} |\phi> where \psi is a gauge fixing fermion, and
where |\phi> is a BRST invariant state determined by simple hermitian
conditions. These state representations are closely related to the path
integral formulation. Here we analyse the basics of this approach in detail.
The freedom in the choice of \psi and |\phi> as well as their properties under
gauge transformations are explicitly determined for simple abelian models. In
all considered cases SL(2,R) is shown both to be a natural extended gauge
symmetry and to be useful to determine |ph>. The results are also applied to
nonabelian models.
