Abstract
Let \({\mathcal H}\) be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for \({\mathcal H}\), which conjecturally puts supercuspidal \({\mathcal H}\)-representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein’s theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for \({\mathcal H}\), that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible \({\mathcal H}\)-representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for \({\mathcal H}\)-representations to the corresponding problem for supercuspidal representations of Levi subgroups of \({\mathcal H}\). The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig’s generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.
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The second author gratefully acknowledges support from the Pacific Institute for the Mathematical Sciences (PIMS). The third author is supported by a NWO Vidi-Grant, No. 639.032.528.
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Aubert, AM., Moussaoui, A. & Solleveld, M. Generalizations of the Springer correspondence and cuspidal Langlands parameters. manuscripta math. 157, 121–192 (2018). https://doi.org/10.1007/s00229-018-1001-8
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DOI: https://doi.org/10.1007/s00229-018-1001-8