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Exotic fusion systems, spetses and counting conjectures Semeraro, Jason


Let G=G(q) be a finite reductive group with Weyl group W. A long time ago Malle noticed that the process by which one calculates the unipotent character degrees of G from the Iwahori-Hecke algebra of W works just as well if W is replaced by a *complex* reflection group (with certain properties). A "spets" is the mysterious object which replaces G in this situation, named after the Greek island Spetses on which these observations were first made. Now suppose l is prime and (l,q)=1. I will argue via the theory of l-compact groups in algebraic topology that we know the l-fusion system of a spets G(q). Moreover, I claim this observation can be combined with results of Cabanes--Enguehard in Deligneâ Lusztig theory to provide degrees of characters in the principal l-block of G. We thus have all the ingredients necessary to formulate exotic counting conjectures inspired by those of Alperin, Dade, Robinson and others. I will explain the proof of such a conjecture when l=2.

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