Webb 猜想与广义 Harish-Chandra 理论
Webb's conjecture and generalised Harish-Chandra theory
AI总结：
本文利用 Deligne–Lusztig 理论构造新单纯复形，将 Webb 猜想推广至有限约化群，并通过广义 Harish-Chandra 条件建立其与轨道空间可缩性及 $e$-尖点对唯一性的联系。
摘要：
Webb 猜想断言，有限群在任意给定素数 $\ell$ 下的 Brown 复形的轨道空间都是可缩的。该猜想由 Symonds 于 1998 年证明。本文提出 Webb 猜想对有限约化群的一个推广。其做法是：借助 Deligne–Lusztig 理论，为每个不可约特征标关联一个新的单纯复形。随后我们证明，所提猜想可由下文称为（$e$-HC-conj）的条件推出，该条件与广义 Harish-Chandra 理论相关。特别地，利用作者此前的结果，我们在对素数 $\ell$ 的温和限制下证明了该猜想，并重新得到适用于有限约化群的 Symonds 结果。最后，我们表明条件（$e$-HC-conj）可由与新定义复形相关的轨道空间的可缩性推出，从而为证明 $e$-尖点对在共轭意义下的唯一性提供了一种尚未探索的拓扑方法。
英文摘要：
Webb's conjecture states that the orbit space of the Brown complex of a finite group at any given prime $\ell$ is contractible. This conjecture was proved by Symonds in 1998. In this paper, we suggest a generalisation of Webb's conjecture for finite reductive groups. This is done by associating to each irreducible character a new simplicial complex defined in terms of Deligne--Lusztig theory. We then show that our conjecture follows from a condition, called ($e$-HC-conj) below, related to generalised Harish-Chandra theory. In particular, using earlier results of the author, we prove our conjecture and recover Symonds result for finite reductive groups under mild restrictions on the prime $\ell$. Finally, we show that the condition ($e$-HC-conj) is implied by the contractibility of the orbit spaces associated to our newly defined complex offering an unexplored topological approach to proving the uniqueness of $e$-cuspidal pairs up to conjugation.
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