Ordinary Weight Conjecture for ell-compact groups

Let $X$ be a simply connected $\ell$-compact group with Weyl group $(W,L)$, with $\ell>2$ very good, and let $\tau$ be an automorphism of $X$ of finite order prime to $\ell$. Let $W\phi=\Phi_X(\tau)$, let $q$ be a power of a prime different from $\ell$, and let $\mathcal F=\mathcal F({{}^{\tau}\!}X(q))$ be the associated fusion system on $S$. For each $s\in S$, let $(W(s),L)$ be the ${\mathbb Z}_\ell$-reflection group underlying $C_X(s)$; let $\zeta\in{\mathbb Z}_\ell^\times$ satisfy $q\equiv\zeta\pmod\ell$; and let $v_s=\nu_\ell(\operatorname{O}_q(W(s)\phi_s))-\nu_\ell(|W(s)_{\phi_s\zeta}|)$. Ordinary Weight Conjecture for ell-compact groups. For every $d\ge0$, one has

$$\mathbf m(\mathcal F,d)=\sum_{s\in S/\mathcal F}|{\operatorname{Irr}}^{d-v_s}(W(s)_{\phi_s\zeta})|.$$

This is a local reformulation and generalisation of the ordinary weight conjecture for spetses and finite groups of Lie type; the paper presents it as conjectural.