Centralisers, complex reflection groups and actions in the Weyl group E6
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
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, .
(doi:10.1007/s40062-023-00326-1).
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.
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Submitted date: 20 December 2021
Accepted/In Press date: 3 April 2023
e-pub ahead of print date: 8 June 2023
Keywords:
Weyl groups, exceptional Lie groups, centralisers, duality, K-theory
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Local EPrints ID: 453270
URI: http://eprints.soton.ac.uk/id/eprint/453270
ISSN: 2193-8407
PURE UUID: 568a55db-2f59-4c72-aef7-aea99676a12a
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Date deposited: 11 Jan 2022 17:51
Last modified: 24 Apr 2024 04:04
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