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The group Diff(S^1) of the orientation preserving diffeomorphisms of thecircle S^1 plays an important role in conformal field theory. We consider asubgroup B_0 of Diff(S^1) whose elements stabilize "the point of infinity".This subgroup is of interest for the actual physical theory living on thepunctured circle, or the real line. We investigate the unique central extensionK of the Lie algebra of that group. We determine the first and secondcohomologies, its ideal structure and the automorphism group. We define ageneralization of Verma modules and determine when these representations areirreducible. Its endomorphism semigroup is investigated and some unitaryrepresentations of the group which do not extend to Diff(S^1) are constructed.

Representation theory of the stabilizer subgroup of the point at infinity in Diff(S^1)