Ordinary Weight Conjecture for spetses

Let G{\mathbb{G}} be a simply connected Z{\mathbb{Z}}_\ell-spets, let >2\ell>2 be very good for G{\mathbb{G}}, and let qq be a power of a prime different from \ell. Let B0B_0 be the principal block of G(q){\mathbb{G}}(q), and let F(G(q))\mathcal F({\mathbb{G}}(q)) be its associated fusion system. Ordinary Weight Conjecture for spetses. For every integer d0d\ge 0, one has

Irrd(B0)=m(F(G(q)),d).|{\operatorname{Irr}}^d(B_0)|=\mathbf m(\mathcal F({\mathbb{G}}(q)),d).

This is an analogue for spetses of Robinson's Ordinary Weight Conjecture; when the spets is associated to a finite group of Lie type, it is equivalent to the ordinary weight conjecture for the principal block of that group.

Sources & referencesView supporting material

Primary source

Radha Kessar, Gunter Malle and Jason Semeraro, “Weight conjectures for -compact groups and spetses”, arXiv:2008.07213 (2023).

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