Weights for compact connected Lie groups (Q6564495)

Let \({\mathbf G}\) be a compact connected Lie group with Weyl group \(W\). Moreover, let \(\ell\) be a prime which is good for \({\mathbf G}\) and let \(S\) be a Sylow \(\ell\)-subgroup of \({\mathbf G}\). Recall that \(S\) is a maximal discrete \(\ell\)-toral subgroup of \({\mathbf G}\), i.e.,\ \(S\) has a normal subgroup \(S^\circ\) such that \(S^\circ\) is isomorphic to a finite product of copies of \({\mathbb Z}/\ell^\infty\) and \(S/S^\circ\) is a finite \(\ell\)-group; moreover, \(S\) is unique up to conjugation. There is a saturated fusion system \({\mathcal F} = {\mathcal F}_S({\mathbf G})\) whose objects are the subgroups of \(S\) and whose morphisms are the group homomorphisms induced by conjugation with elements in \({\mathbf G}\). It is known that \(S\) has only finitely many \({\mathcal F}\)-classes of \({\mathcal F}\)-centric \({\mathcal F}\)-radical subgroups. Let \(\mathfrak Q\) be a set of representatives for these. The number of weights of \({\mathcal F}\) is \({\mathbf w}({\mathcal F}) := \sum_{Q \in {\mathfrak Q}} z(\mathrm{Out}_{\mathcal F}(Q))\) where \(z(\mathrm{Out}_{\mathcal F}(Q))\) denotes the number of irreducible characters of \(\ell\)-defect zero of the finite group \(\mathrm{Out}_{\mathcal F}(Q) := \Aut_{\mathcal F}(Q)/\mathrm{Inn}(Q)\). The authors prove that \(\mathbf{w}(\mathcal{F})= |\mathrm{Irr}(W)|\); this can be seen as a version of Alperin's weight conjecture in an infinite setting. From this result, the authors deduce a similar fact for connected reductive algebraic groups over \(\overline{\mathbb{F}}_p\) where \(p \neq \ell\) is a prime. They also give examples, for bad primes \(\ell\), where \(\mathbf{w}(\mathcal{F}) \neq |\mathrm{Irr} (W)|\).