REPRESENTATION THEORY OF THE STABILIZER SUBGROUP OF THE POINT AT INFINITY IN Diff(S1)

Abstract

The group Diff (S1) of the orientation-preserving diffeomorphisms of the circle S1 plays an important role in conformal field theory. We consider a subgroup B0 of Diff (S1) whose elements stabilize "the point at infinity." This subgroup is of interest for the actual physical theory that lives on the punctured circle, or the real line. We investigate the unique central extension [Formula: see text] of the Lie algebra of that group. We determine the first and second cohomologies, its ideal structure and the automorphism group. We define a generalization of Verma modules and determine when these representations are irreducible. Its endomorphism semigroup is investigated and some unitary representations of the group which do not extend to Diff (S1) are constructed.