  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.
