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

TL;DR

This paper provides a topological framework for understanding highest weight representations of Kac-Moody groups using equivariant K-theory, connecting algebraic representations with topological invariants, especially for groups of compact and extended compact types.

Contribution

It introduces a topological interpretation of highest weight representations via equivariant K-theory and computes these groups for specific Kac-Moody groups, including E_{10}.

Findings

Grothendieck group of integrable highest weight representations maps isomorphically to equivariant K-theory.

Explicit computation of K_G^*(EG) for Kac-Moody groups of extended compact type.

Alignment with recent results of Freed-Hopkins-Teleman in the affine case.

Abstract

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}.