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

TL;DR

This paper explores the homotopy properties of maximal tori in the $E_6$ Lie group, showing duality relations, decomposing centralisers, and computing related $K$-theory, extending previous results from simpler cases.

Contribution

It computes extended quotients of maximal tori for $E_6$, proves homotopy equivalences in dual sectors, and generalizes results on centralisers and component groups.

Findings

Homotopy equivalences of sectors in $E_6$ are established.

Centralisers in the $E_6$ Weyl group decompose as products of reflection groups.

The $K$-theory of the Iwahori-spherical $C^*$-algebra for $E_6$ is computed.

Abstract

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.