Centralisers, complex reflection groups and actions in the Weyl group $$E_6$$

Abstract

AbstractThe compact, connected Lie group $$E_6$$ 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 $$A_n$$ 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 $$E_6$$ E 6 , showing that the homotopy equivalences of sectors established in the $$A_n$$ 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 $$E_6$$ 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 $$C^*$$ C ∗ -algebra of the p-adic group $$E_6$$ E 6 , which may be of adjoint type or simply connected.