Dominant K-theory and integrable highest weight representations of Kac–Moody groups

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, KG 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 highest weight representations of a Kac–Moody group G of compact type, maps isomorphically onto , 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 for Kac–Moody groups of extended compact type, which includes the Kac–Moody group E10.