We consider the unicity of types for various classes of supercuspidal representations of reductive p-adic groups, with a view towards establishing instances of the inertial Langlands correspondence. We introduce the notion of an archetype, which we define to be a conjugacy class of typical representations of maximal compact subgroups. In the case of supercuspidal representations of a special linear group, we generalize the functorial results of Bushnell and Kutzko relating simple types in GLN(F) and SLN(F) to cover all archetypes; from this we deduce that any archetype for a supercuspidal representation of SLN(F) is induced from a maximal simple type. We then provide an explicit description of the number of archetypes contained in a given supercuspidal representation of SLN(F). We next consider depth zero supercuspidal representations of an arbitrary group, where we are able to show that theonly archetypes are the depth zero types constructed by Morris. We end by showing that there exists a unique inertial Langlands correspondence from the set of archetypes contained in regular supercuspidal representations to the set of regular inertial types. In the cases of SLN(F) and depth zero supercuspidals of arbitrary groups, we describe completely the fibres of this inertial correspondence; in general, we formulate a conjecture on how these fibres should look for all regular inertial types.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:695039 |
| Date | January 2016 |
| Creators | Latham, Peter |
| Publisher | University of East Anglia |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ueaeprints.uea.ac.uk/60655/ |
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