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In 1993, Brou\'{e}, Malle and Michel initiated the study of spetses on theGreek island bearing the same name. These are mysterious objects attached tonon-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 Liegroup. A connected $p$-compact group $\mathbf{X}$ is determined by its rootdatum which in turn determines its Weyl group $W_\mathbf{X}$. In this articlewe give strong numerical evidence for a connection between these two objects byconsidering the case when $\mathbf{X}$ is the exotic $2$-compact group DI$(4)$constructed by Dwyer--Wilkerson and $W_\mathbf{X}$ is the complex reflectiongroup $G_{24} \cong$ GL$_3(2) \times C_2$. Inspired by results inDeligne--Lusztig theory for classical groups, if $q$ is an odd prime power wepropose a set Irr$(\mathbf{X}(q))$ of `ordinary irreducible characters'associated to the space $\mathbf{X}(q)$ of homotopy fixed points under theunstable Adams operation $\psi^q$. Notably Irr$(\mathbf{X}(q))$ includes theset of unipotent characters associated to $G_{24}$ constructed by Brou\'{e},Malle and Michel from the Hecke algebra of $G_{24}$ using the theory ofspetses. By regarding $\mathbf{X}(q)$ as the classifying space of aBenson--Solomon fusion system Sol$(q)$ we formulate and prove an analogue ofRobinson's ordinary weight conjecture that the number of characters of defect$d$ in Irr$(\mathbf{X}(q))$ can be counted locally.

A $2$-compact group as a spets