  Let K be a connected compact Lie group, and G be its complexification. The
homology of the based loop group \Omega K with integer coefficients is
naturally a \ZZ-Hopf algebra. After possibly inverting 2 or 3, we identify
H_*(\Omega K,\ZZ) 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\Lie B^\vee. We also give a similar interpretation for the equivariant
homology of \Omega K under the maximal torus action.
