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- W4280490114 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." @default.
- W4280490114 created "2022-05-22" @default.
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- W4280490114 date "2022-05-17" @default.
- W4280490114 modified "2023-09-30" @default.
- W4280490114 title "Centralisers, complex reflection groups and actions in the Weyl group $E_6$" @default.
- W4280490114 doi "https://doi.org/10.48550/arxiv.2205.08230" @default.
- W4280490114 hasPublicationYear "2022" @default.
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