  We give a topological interpretation of the highest weight representations of
Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we
define a version of equivariant K-theory, K_G on the category of proper G-CW
complexes. We then study Kac-Moody groups of compact type in detail (see
Section 2 for definitions). In particular, we show that the Grothendieck group
of integrable hightest weight representations of a Kac-Moody group G of compact
type, maps isomorphically onto K_G^*(EG), where $EG$ is the classifying space
of proper G-actions. For the affine case, this agrees very well with recent
results of Freed-Hopkins-Teleman. We also explicitly compute K_G^*(EG) for
Kac-Moody groups of extended compact type, which includes the Kac-Moody group
E_{10}.
