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A stacky generalized Springer correspondence and rigid enhancements of L-parameters
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Motivated by applications to the Langlands program, Aubert-Moussaoui-Solleveld extended Lusztig's generalized Springer correspondence to disconnected reductive groups. We use stacks to give a more geometric account of their theory, in particular, formulating a truly geometric version of the (relevant analogue of the) Bernstein-Zelevinsky Geometrical Lemma and explaining how to compare the correspondence on the group and the Lie algebra using quasi-logarithms. As an application, we study Kaletha's rigid enhancements of L-parameters and draw the same conclusions as Aubert-Moussaoui-Solleveld for this enhancement: there exists a cuspidal support map and its fibers are parameterized by irreducible representations of twisted group algebras.
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The Aubert and Bernstein involutions for disconnected groups
Aubert and Bernstein dualities extend to disconnected reductive p-adic groups, with uniqueness, irreducibility preservation, character formulas, and a Steinberg representation for twisted endoscopy.
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