Centralisers, complex reflection groups and actions in the Weyl group \(E_6\)

The compact, connected Lie group \(E_6\) admits two forms: simply connected and adjoint type. As we previously established, the Baum–Connes isomorphism relates the two Langlands dual forms, giving a duality between the equivariant K-theory of the Weyl group acting on the corresponding maximal tori. Our study of the \(A_n\) case showed that this duality persists at the level of homotopy, not just homology. In this paper we compute the extended quotients of maximal tori for the two forms of \(E_6\), showing that the homotopy equivalences of sectors established in the \(A_n\) case also exist here, leading to a conjecture that the homotopy equivalences always exist for Langlands dual pairs. In computing these sectors we show that centralisers in the \(E_6\) Weyl group decompose as direct products of reflection groups, generalising Springer’s results for regular elements, and we develop a pairing between the component groups of fixed sets generalising Reeder’s results. As a further application we compute the K-theory of the reduced Iwahori-spherical \(C^*\)-algebra of the p-adic group \(E_6\), which may be of adjoint type or simply connected.