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Original Articles

A 2-Compact Group as a Spets

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Pages 140-155 | Published online: 08 Jul 2021
 

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 WX. 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 DI(4) constructed by Dwyer–Wilkerson and WX is the complex reflection group G24GL3(2)×C2. Inspired by results in Deligne–Lusztig theory for classical groups, if q is an odd prime power, then we propose a set Irr(X(q)) of “ordinary irreducible characters” associated to the space X(q) of homotopy fixed points under the unstable Adams operation ψq. Notably, Irr(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 X(q) as the classifying space of a Benson–Solomon fusion system Sol(q), we formulate and prove an analogue of Robinson’s ordinary weight conjecture that the number of characters of defect d in Irr(X(q)) can be counted locally.

2010 Mathematics Subject Classification:

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 and 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).

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