  The purpose of the present paper is to investigate the necessary conditions
for unitarity of the spectrum of non-compact gauged WZNW models to some depth.
In particular, we would like to investigate the necessity of integer weights
and level. We will learn that the problem is very complex and we have not found
any simple and general way to formulate the necessary conditions. Instead one
must resort to studying the problem almost case by case. The only nearly
complete conditions that we will find, is for the case g = su(n,1).
Furthermore, the horizontal part of the case g = su(p,q) is nearly completed as
well. In other cases, we will find conditions associated with certain
subalgebras and nodes in the Dynkin diagram close to the one corresponding to
the non-compact root. In these examples we can give conditions for the
horizontal part of the algebra. As a by-product of our investigation we will
prove some nice formulae of character identities and explicit branching
functions for these representations.
