A pinned canonical Jordan decomposition gives a bijection from Lusztig series to unipotent characters for disconnected dual centralizers and induces canonical isomorphisms between depth-zero Bernstein Hecke algebras and unipotent Hecke algebras.
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Pinned Jordan Decomposition of Characters and Depth-Zero Hecke Algebras
A pinned canonical Jordan decomposition gives a bijection from Lusztig series to unipotent characters for disconnected dual centralizers and induces canonical isomorphisms between depth-zero Bernstein Hecke algebras and unipotent Hecke algebras.