Multiplets of Representations and Kostant’s Dirac Operator for Equal Rank Loop Groups

Let g be a semisimple Lie algebra, and let h be a reductive subalgebra of maximal rank in g. Given any irreducible representation of g, consider its tensor product with the spin representation associated to the orthogonal complement of h in g. Recently, B. Gross, B. Kostant, P. Ramond, and S. Sternberg [2] proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of h, called a multiplet. Kostant [7] then constructed a formal h-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of h-representations corresponding to the given representation of g. We reproduce these results in the Kac-Moody setting for the extended loop algebras L̃g and L̃h. We prove a homogeneous generalization of the Weyl-Kac character formula, which now yields a multiplet of irreducible positive energy representations of Lh associated to any irreducible positive energy representation of Lg. We construct an Lh-equivariant operator, analogous to Kostant’s Dirac operator, on the tensor product of a representation of Lg with the spin representation associated to the complement of Lh in Lg. We then prove that the kernel of this operator gives the Lh-multiplet corresponding to the original representation of Lg.