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 finite C oxeter groups and I wahori- H ecke algebras , volume 21 of London Mathematical Society Monographs
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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.