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