Finite-field reductive group representation series with fixed semisimple parameter are canonically equivalent to unipotent representations of endoscopic groups with equivariant structure, including for disconnected groups.
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Endoscopy for representations of disconnected reductive groups over finite fields
Finite-field reductive group representation series with fixed semisimple parameter are canonically equivalent to unipotent representations of endoscopic groups with equivariant structure, including for disconnected groups.