A 2-Compact Group as a Spets

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.

Keywords:

• fusion system
• block
• p-compact group
• spetses

2010 Mathematics Subject Classification:

• 20C20
• 20D20
• 20D06
• 55R35

Acknowledgments

The author thanks Radha Kessar, Jesper Grodal and Gunter Malle for their comments on earlier versions of this article. Thanks also to Frank Lübeck for providing Tables 8 and 9 and to Jay Taylor for helping me understand them. I am also grateful to David Craven for suggesting the main approach used in the proof of Theorem 2.2, and to Markus Linckelmann and Justin Lynd for helpful conversations. Finally, I thank to the Mathematisches Forschungsinstitut Oberwolfach for its hospitality during the week-long workshop “Representations of Finite Groups” in March 2019. It was there that many of the ideas in this article were conceived. Finally, I thank to the anonymous referees for their careful reading and suggestions which have led to numerous improvements.

Declaration of Interest

No potential conflict of interest was reported by the author(s).