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Fundamental conjectures in modular representation theory of finite groups,more precisely, Alperin's Weight Conjecture and Robinson's Ordinary WeightConjecture, can be expressed in terms of fusion systems. We use fusion systemsto connect the modular representation theory of finite groups of Lie type tothe theory of $\ell$-compact groups. Under some mild conditions we prove thatthe fusion systems associated to homotopy fixed points of $\ell$-compact groupssatisfy an equation which for finite groups of Lie type is equivalent toAlperin's Weight Conjecture. For finite reductive groups, Robinson's OrdinaryWeight Conjecture is closely related to Lusztig's Jordan decomposition ofcharacters and the corresponding results for Brauer $\ell$-blocks. Motivated bythis, we define the principal block of a spets attached to a spetsial ${\mathbbZ}_\ell$-reflection group, using the fusion system related to it via$\ell$-compact groups, and formulate an analogue of Robinson's conjecture forthis block. We prove this formulation for an infinite family of cases as wellas for some groups of exceptional type. Our results not only provide furtherstrong evidence for the validity of the weight conjectures, but also pointtoward some yet unknown structural explanation for them purely in the frameworkof fusion systems.

Weight conjectures for $\ell$-compact groups and spetses