论文标题
Centeriser，复杂的反射组和Weyl Group $ e_6 $中的动作
Centralisers, complex reflection groups and actions in the Weyl group $E_6$
论文作者
论文摘要
紧凑的，连接的谎言组$ e_6 $允许两种形式：简单地连接和伴随类型。正如我们先前建立的那样，鲍姆 - 康涅狄格部同构象征性关系是两种兰兰双重形式，在作用于相应的最大托里的Weyl基理论之间具有双重性。我们对$ a_n $案例的研究表明，这种二元性持续在同质级别上，而不仅仅是同源性。在本文中，我们计算了两种形式的$ e_6 $的最大托里的扩展商，这表明这里还存在$ a_n $案例中建立的扇区的同质等效，从而导致猜想，即兰兰斯双对始终存在同型等效性。在计算这些扇区时，我们表明$ e_6 $ weyl组中的centerisers将作为反射组的直接产物分解为常规元素的春季结果，并且我们在固定集的组件组之间开发了配对，从而推广了Reeder的结果。作为进一步的应用程序，我们计算了$ k $ - 降低的iwahori-spherical $ c^*$ - P-ADIC组的代数$ e_6 $，这可能是相邻类型或简单地连接的。
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.
