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We construct projective unitary representations of (a) Map^ G), the group of smooth maps from the circle into a compact Lie group G, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of MapίS 1 T), where T is a maximal torus of G, can be extended to representations of MapCS^G),1 To avoid misunderstanding I should emphasize that in this paper the irreducibility of P q2 is proved only when q = 0 (cf. Sect. 6). The irreducibility has been proved in general by Kac
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