Integral homology of loop groups via Langlands dual groups

Let K be a connected compact Lie group, and G its complexification. The homology of the based loop group ΩK with integer coefficients is naturally a Z-Hopf algebra. After possibly inverting 2 or 3, we identify H∗(ΩK, Z) with the Hopf algebra of algebraic functions on B∨ e , where B^∨ is a Borel subgroup of the Langlands dual group scheme G^∨ of G and B^∨ e is the centralizer in B^∨ of a regular nilpotent element e ∈ LieB^∨. We also give a similar interpretation for the equivariant homology of ΩK under the maximal torus action.