In Deligne-Lusztig theory we take an alternating sum over cohomology in all degrees. Given an irrep of a finite group of Lie type can we trace back which degree it shows up in?
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1$\begingroup$ For a general irrep, no. E.g. the Steinberg rep of SL_2(F_q) can appear in both H^1 and H^2, depending on which torus you are using. Maybe you should restrict to the cuspidal ones. But then "Given an irrep" seems need some explanation (e.g. do you mean given the set of character values of a character?) $\endgroup$– user148212Sep 19, 2021 at 0:21
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