Centralisers, complex reflection groups and actions in the Weyl group E6

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.

Weyl groups, exceptional Lie groups, centralisers, duality, K-theory

10.1007/s40062-023-00326-1

2193-8407

219–264

Wright, Nicholas

f4685b8d-7496-47dc-95f0-aba3f70fbccd

Niblo, Graham

43fe9561-c483-4cdf-bee5-0de388b78944

Plymen, Roger

76de3dd0-ddcb-4a34-98e1-257dddb731f5

Wright, Nicholas

f4685b8d-7496-47dc-95f0-aba3f70fbccd

Niblo, Graham

43fe9561-c483-4cdf-bee5-0de388b78944

Plymen, Roger

76de3dd0-ddcb-4a34-98e1-257dddb731f5

Wright, Nicholas, Niblo, Graham and Plymen, Roger (2023) Centralisers, complex reflection groups and actions in the Weyl group E6. Journal of Homotopy and Related Structures, 18, 219–264. (doi:10.1007/s40062-023-00326-1).

Record type: Article

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.