Counting conjectures and e-local structures in finite reductive groups
Damiano Rossi
Abstract
We prove new results in generalised Harish-Chandra theory providing a description of the so-called Brauer–Lusztig blocks in terms of 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 parametrisations of generalised Harish-Chandra series.