Counting conjectures and e-local structures in finite reductive groups

2022/04/01 by Rossi, Damiano · 1 citation
#20C20 #20C33 (20G40) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2204.00428

Abstract

We prove new results in generalized Harish-Chandra theory providing a description of the so-called Brauer--Lusztig blocks in terms of the information encoded in the ℓ-adic cohomology of Deligne--Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the ℓ-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Broué, Fong and Srinivasan between ℓ-structures and their geometric counterpart. Finally, using the description of Brauer--Lusztig blocks mentioned above, we reduce our conjectures to the verification of Clifford theoretic properties expected from certain parametrisation of generalised Harish-Chandra series.

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• Generic weights for finite reductive groups

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