Centralisers, complex reflection groups and actions in the Weyl group $E_6$
| Autor: | Niblo, Graham A., Plymen, Roger, Wright, Nick |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s40062-023-00326-1 |
| Popis: | 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. Comment: J. Homotopy Relat. Struct. (2023) |
| Databáze: | arXiv |
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