On the twisted K-homology of simple Lie groups

We prove that the twisted $K$-homology of a simply connected simple Lie group $G$ of rank $n$ is an exterior algebra on $n-1$ generators tensor a cyclic group. We give a detailed description of the order of this cyclic group in terms of the dimensions of irreducible representations of $G$ and show that the congruences determining this cyclic order lift along the twisted index map to relations in the twisted ${\mathrm{Spin}}^c$ bordism group of $G$.