Weight conjectures for \(\ell\)-compact groups and spetses (Q6596067)

In this important paper, the authors connect the representation theory of finite groups of Lie type with \(\ell\)-compect groups and spets by way of fusion systems, giving analogues of Alperin's Weight Conjecture (AWC) and Robinson's Ordinary Weight Conjecture (OWC).\N\NAlperin's Weight Conjecture for the principal \(l\)-block \(B_0\) of a finite group \(G\) states that the number \(|\mathrm{IBr}(B_0)|\) of irreducible Brauer characters is equal to a weight expression \(\mathbf{w}(\mathcal{F})\) derived from the fusion system \(\mathcal{F}\) of \(G\) on a Sylow \(p\)-subgroup. When \(G=\mathbf{G}^F\) is a finite group of Lie type defined over a prime power \(q\) of order \(e\) modulo \(\ell\), the authors prove that \(|\mathrm{IBr}(B_0)=|\mathrm{Irr}(W_e)|\), where \(W_e\) is the relative Weyl group, so that the statement of AWC in this case becomes \(\mathbf{w}(\mathcal{F})=|\mathrm{Irr}(W_e)|\), and it is this expression that is the basis for what follows.\N\NAn \(\ell\)-compact group \(X\) has an associated reflection group \(W\), and if \(X\) is simply connected and \(\ell\) is odd, then given a self equivalence \(\tau\) of \(X\) and a prime power \(q\) not divisible by \(\ell\), there is a fusion system \(\mathcal{F}({}^\tau X(q))\). In the case that \(W\) is rational with associated finite group of Lie type \(\mathbf{G}^F\) defined over \(q\), there is \(\ell\)-compact \(X\) and \(\tau\) such that the fusion system for \(\mathbf{G}^F\) is \(\mathcal{F}({}^\tau X(q))\). This leads to a statement of AWC for simply connected \(\ell\)-compact groups for \(\ell\) odd, that \(\mathbf{w}(\mathcal{F}({}^\tau X(q)))=|\mathrm{Irr}(W_e)|\), shown to be true when \(\ell\) is a \textit{very good prime}. As a consequence this gives some new cases for AWC for simply connected groups of Lie type for very good primes, in particular types \(E_6\), \(E_7\) and \(E_8\).\N\NNow consider \(W\) a spetsial Weyl group, with associated \(\mathbb{Z}_{\ell}\)-spets \(\mathbb{G}\). For \(\ell\) very good, the authors define the principal \(\ell\)-block \(\mathrm{Irr}(B_0)\) for \(\mathbb{G}\). Using an \(\ell\)-compact group associated to \(W\), the authors define a fusion system for \(\mathbb{G}(q)\), from which can be defined a weight expression as in the OWC. Together with the definition of \(\mathrm{Irr}(B_0)\), this gives the statement of an OWC for spetses. This conjecture is then generalized to arbitrary simply connected \(\ell\)-compact groups. As evidence, the conjecture is proved when \(\ell\) does not divide \(|W|\), and for the smallest non-trivial infinite family of reflection groups.