  Since the work of Bershadsky and Ooguri and Feigin and Frenkel it is well
known that correlators of $SL(2)$ current algebra for admissible
representations should reduce to correlators for conformal minimal models. A
precise proposal for this relation has been given at the level of correlators:
When $SL(2)$ primary fields are expressed as $\phi_j(z_n,x_n)$ with $x_n$ being
a variable to keep track of the $SL(2)$ representation multiplet (possibly
infinitely dimensional for admissible representations), then the minimal model
correlator is supposed to be obtained simply by putting all $x_n=z_n$. Although
strong support for this has been presented, to the best of our understanding a
direct, simple proof seems to be missing so in this paper we present one based
on the free field Wakimoto construction and our previous study of that in the
present context. We further verify that the explicit $SL(2)$ correlators we
have published in a recent preprint reduce in the above way, up to a constant
which we also calculate. We further discuss the relation to more standard
formulations of hamiltonian reduction.
