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

TL;DR

This paper establishes a connection between Poincaré duality and Langlands duality for extended affine Weyl groups, showing how dual tori with group actions relate through K-theory isomorphisms.

Contribution

It constructs an equivariant Poincaré duality framework and demonstrates that Langlands duality induces a rational K-theory isomorphism between extended affine Weyl group C*-algebras.

Findings

Established equivariant Poincaré duality between dual tori with finite group actions

Proved Langlands duality induces a rational isomorphism at the K-theory level

Connected duality theories through C*-algebra K-theory analysis

Abstract

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 $C^*$-algebras of extended affine Weyl groups at the level of $K$-theory.