A 2-Compact Group as a Spets

Abstract

In 1998 Malle introduced spetses which are mysterious objects with 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 subjects by considering the case when $\\mathbf{X}$ is the exotic $2$-compact group $\\operatorname{DI}(4)$ constructed by Dwyer--Wilkerson and $W_\\mathbf{X}$ is the complex reflection group $G_{24} \\cong \\operatorname{GL}_3(2) \\times C_2$. Inspired by results in Deligne--Lusztig theory for classical groups, if $q$ is an odd prime power we propose a set $\\operatorname{Irr}(\\mathbf{X}(q))$ of `ordinary irreducible characters associated to the space $\\mathbf{X}(q)$ of homotopy fixed points under the unstable Adams operation $\\psi^q$. Notably $\\operatorname{Irr}(\\mathbf{X}(q))$ includes the set of unipotent characters associated to $G_{24}$ constructed by Broue, 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 $\\operatorname{Sol}(q)$ we formulate an analogue of Robinson s ordinary weight conjecture that the number of characters of defect $d$ in $\\operatorname{Irr}(\\mathbf{X}(q))$ can be counted locally, and prove this when $q=3$. We also consider some similar conjectures at odd primes.