A pinned canonical bijection is constructed between Lusztig series and unipotent characters of possibly disconnected dual centralizers for finite reductive groups, with an enriched version for disconnected groups.
Representations of Reductive Groups Over Finite Fields , urldate =
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Pinned Jordan Decomposition of Characters and Depth-Zero Hecke Algebras
A pinned canonical bijection is constructed between Lusztig series and unipotent characters of possibly disconnected dual centralizers for finite reductive groups, with an enriched version for disconnected groups.