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. Furthermore, we apply these results to obtain new progress towards the verification of the inductive condition for Dade's Conjecture in the case of groups of Lie type in non-defining characteristic.
On the Cohomology of Deligne-Lusztig Varieties
The projective cover of the trivial representation for a finite group of Lie type in defining characteristic
Classifying representations of finite classical groups of Lie type of dimension up to $\ell^4$
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