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Previously we gave a proof of the Feigin--Fuchs character formula for theirreducible unitary discrete series of the Virasoro algebra with 0<c<1. Theproof showed directly that the mutliplicity space arising in the cosetconstruction of Goddard, Kent and Olive was irreducible, using the elementarypart of the unitarity criterion of Friedan, Qiu and Shenker, givingrestrictions on h for c=1-6/m(m+1) with m>2. In this paper we consider the sameproblem in the limiting case of the coset construction c=1. Using primaryfields, we directly establish that the Virasoro algebra acts irreducibly on themultiplicity spaces of irreducible representations of SU(2) in the two levelone irreducible representations of the corresponding affine Kac--Moody algebra.This gives a direct proof that the only singular vectors in theserepresentations are those given by Goldstone's formulas, which also play animportant part in the proof. For this proof, the theory is developed fromscratch in a self-contained semi-expository way. Using the Jantzen filtrationand the Kac determinant formula, we give an additional independent proof forthe case c=1 which generalises to the case 0<c<1, where it provides analternative approach to that of Astashkevich.

Direct proofs of the Feigin-Fuchs character formula for unitary representations of the Virasoro algebra