Poincar\'e duality and Langlands duality for extended affine Weyl groups

In this paper we construct an equivariant Poincar'e duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group $C^{\ast }$-algebras of extended affine Weyl groups at the level of $K$-theory.