Twisted Heisenberg Central Extensions and the Affine ADE Basic Representation

We study various aspects of the representation theory of loop groups, all with the aim of giving geometric constructions, parameterized by conjugacy classes of the Weyl group, of the basic representation of the affine Lie algebras associated to a simply laced simple Lie algebra as a restriction isomorphism on dual sections of the level 1 line bundle on the affine Grassmannian. Along the way, we obtain various results on the structure of loop tori, the definition of a notion of a Heisenberg Central extension as an alternative for twisted modules over the lattice vertex algebra and the determination of their representation theory, some computations on central extensions of a torus over a field by K 2 , and a new proof of the classification of the conjugacy classes of the Weyl group by parabolic induction.