A 2-Compact Group as a Spets. Issue 1 (2nd January 2023) - Semeraro, Jason
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 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 DI ( 4 ) constructed by Dwyer–Wilkerson and W X is the complex reflection group G 24 ≅ GL 3 ( 2 ) × C 2 . 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.