A $2$-compact group as a spets

Abstract

In 1993, Brou\'{e}, 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 $\mathbf{X}$ is a space which is a homotopy-theoretic $p$-local analogue of a compact Lie group. A connected $p$-compact group $\mathbf{X}$ 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 $\mathbf{X}$ is the exotic $2$-compact group DI$(4)$ constructed by Dwyer--Wilkerson and $W_\mathbf{X}$ is the complex reflection group $G_{24} \cong$ GL$_3(2) \times C_2$. Inspired by results in Deligne--Lusztig theory for classical groups, if $q$ is an odd prime power we propose a set Irr$(\mathbf{X}(q))$ of `ordinary irreducible characters' associated to the space $\mathbf{X}(q)$ of homotopy fixed points under the unstable Adams operation $\psi^q$. Notably Irr$(\mathbf{X}(q))$ includes the set of unipotent characters associated to $G_{24}$ constructed by Brou\'{e}, Malle and Michel from the Hecke algebra of $G_{24}$ using the theory of spetses. By regarding $\mathbf{X}(q)$ 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$(\mathbf{X}(q))$ can be counted locally.