Integral homology of loop groups via Langlands dual groups

Abstract

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