  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.
