Counting conjectures and $e$-local structures in finite reductive groups

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 $\ell$-adic cohomology of Deligne--Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the $\ell$-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 Brou\'e, Fong and Srinivasan between $\ell$-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.