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

TL;DR

This paper studies the structure and representations of a subgroup of Diff(S^1) that stabilizes a point at infinity, with implications for conformal field theory on punctured circles.

Contribution

It characterizes the cohomology, automorphisms, and representations of the stabilizer subgroup of Diff(S^1), including a generalization of Verma modules and new unitary representations.

Findings

Determined the first and second cohomologies of the subgroup.

Classified irreducible generalized Verma modules.

Constructed new unitary representations not extending to Diff(S^1).

Abstract

The group Diff(S^1) of the orientation preserving diffeomorphisms of the circle S^1 plays an important role in conformal field theory. We consider a subgroup 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 the punctured circle, or the real line. We investigate the unique central extension K 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(S^1) are constructed.