Let $\mathfrak {g}$ be a semisimple Lie algebra, and let $\mathfrak
{h}$ be a reductive subalgebra of maximal rank in $\mathfrak
{g}$. Given any irreducible representation of $\mathfrak {g}$,
consider its tensor product with the spin representation associated to
the orthogonal complement of $\mathfrak {h}$ in $\mathfrak
{g}$. 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 $\mathfrak {h}$,
called a multiplet. Kostant [7] then constructed a formal $\mathfrak
{h}$-equivariant Dirac operator on such product representations whose
kernel is precisely the multiplet of $\mathfrak {h}$-representations
corresponding to the given representation of $\mathfrak {g}$.
¶ We reproduce these results in the Kac-Moody setting for the
extended loop algebras $\tilde {L}\mathfrak {g}$ and $\tilde
{L}\mathfrak {h}$. We prove a homogeneous generalization of the
Weyl-Kac character formula, which now yields a multiplet of
irreducible positive energy representations of $L\mathfrak {h}$
associated to any irreducible positive energy representation of
$L\mathfrak {g}$. We construct an $L\mathfrak {h}$-equivariant
operator, analogous to Kostant's Dirac operator, on the tensor product
of a representation of $L\mathfrak {g}$ with the spin representation
associated to the complement of $L\mathfrak {h}$ in $L\mathfrak
{g}$. We then prove that the kernel of this operator gives the
$L\mathfrak {h}$-multiplet corresponding to the original
representation of $L\mathfrak {g}$.