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 $\Omega K$ with integer coefficients is naturally a $\mathbb{Z}$-Hopf algebra. After possibly inverting $2$ or $3$, we identify $H_*(\Omega K,\mathbb{Z})$ with the Hopf algebra of algebraic functions on $B^\vee_e$, where $B^\vee$ is a Borel subgroup of the Langlands dual group scheme $G^\vee$ of $G$ and $B^\vee_e$ is the centralizer in $B^\vee$ of a regular nilpotent element $e\in\operatorname{Lie} B^\vee$. We also give a similar interpretation for the equivariant homology of $\Omega K$ under the maximal torus action.