  We study the cohomology arising in the BRST formulation of G/H gauged WZNW
models, i.e. in which the states of the gauged theory are projected out from
the ungauged one by means of a BRST condition. We will derive for a general
simple group $H$ with arbitrary level, conditions for which the cohomology is
non-trivial. We show, by introducing a small perturbation due to Jantzen, in
the highest weights of the representations, how states in the cohomology,
"singlet pairs", arise from unphysical states, "Kugo-Ojima quartets", as the
perturbation is set to zero. This will enable us to identify and construct
states in the cohomology. The ghost numbers that will occur are $\pm p$, with
$p$ uniquely determined by the representations of the algebras involved. Our
construction is given in terms of the current modes and relies on the explicit
form of highest weight null-states given by Malikov, Feigen and Fuchs.
