Jordan decomposition commutes with Harish-Chandra induction for all finite connected reductive groups, via an auxiliary result on endomorphism algebras of induced cuspidals.
Characters of reductive groups over a finite field , volume 107 of Annals of Mathematics Studies
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Harish-Chandra Induction and Jordan Decomposition of Characters
Jordan decomposition commutes with Harish-Chandra induction for all finite connected reductive groups, via an auxiliary result on endomorphism algebras of induced cuspidals.