Abstract

In 1993, Broué, Malle and Michel initiated the study of spetses on the Greek island bearing the same name. These are mysterious objects attached to non-real Weyl groups. In algebraic topology, a p-compact group X is a space which is a homotopy-theoretic p-local analogue of a compact Lie group. A connected p-compact group X is determined by its root datum which in turn determines its Weyl group $W_{\mathbf{X}}$. In this article, we give strong numerical evidence for a connection between these two objects by considering the case when X is the exotic 2-compact group $\text{DI}(4)$ constructed by Dwyer–Wilkerson and $W_{\mathbf{X}}$ is the complex reflection group $G_{24}\cong {\text{GL}}_3(2)\times C_2$. Inspired by results in Deligne–Lusztig theory for classical groups, if q is an odd prime power, then we propose a set $\text{Irr}(\mathbf{X}(q))$ of “ordinary irreducible characters” associated to the space $\mathbf{X}(q)$ of homotopy fixed points under the unstable Adams operation ψq. Notably, $\text{Irr}(\mathbf{X}(q))$ includes the set of unipotent characters associated to G24 constructed by Broué, Malle and Michel from the Hecke algebra of G24 using the theory of spetses. By regarding $\mathbf{X}(q)$ as the classifying space of a Benson–Solomon fusion system $\text{Sol}(q),$ we formulate and prove an analogue of Robinson’s ordinary weight conjecture that the number of characters of defect d in $\text{Irr}(\mathbf{X}(q))$ can be counted locally.

摘要

1993 年，Broué、Malle 和 Michel 发起了对希腊同名岛屿上斯派赛斯岛的研究。这些是附属于非真实外尔群的神秘物体。在代数拓扑中，p-紧群 X 是一个空间，它是紧李群的同伦论 p-局部类比。连通的 p-紧群 X 由其根基准决定，根基准又决定其 Weyl 群$W_{\mathbf{X}}$. 在本文中，我们通过考虑 X 是奇异 2-紧群的情况，给出了这两个对象之间联系的强有力的数值证据$\text{DI}(\mathrm{4个})$由 Dwyer-Wilkerson 和$W_{\mathbf{X}}$是复反射群$G_{24}\cong {\text{GL}}_{\mathrm{3个}}(\mathrm{2个})\times C_{\mathrm{2个}}$. 受经典群的 Deligne–Lusztig 理论结果的启发，如果 q 是奇素数幂，则我们提出一个集合$\text{红外线}(\mathbf{X}(q))$与空间相关的“普通不可约字符”$\mathbf{X}(q)$不稳定 Adams 运算 ψq 下的同伦不动点数。尤其，$\text{红外线}(\mathbf{X}(q))$包括由 Broué、Malle 和 Michel 使用 spetses 理论从 G24 的 Hecke 代数构造的与 G24 相关的单能特征集。通过考虑$\mathbf{X}(q)$作为 Benson-Solomon 融合系统的分类空间$\text{溶胶}(q),$我们制定并证明了罗宾逊的普通权重猜想的类比，即缺陷 d 中的字符数$\text{红外线}(\mathbf{X}(q))$可以在当地算。