Abstract
Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.
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Acknowledgements
The authors wish to thank Vincent Sécherre for some helpful explanations on the construction of simple types for GL m (D), Mark Reeder for pointing out some examples where the depth is not preserved, Wilhelm Zink for explaining properties of the Artin reciprocity map and Paul Broussous for providing the reference [BaBr], which has allowed a substantial simplification of the proof of Proposition 2.6 from a previous version. The second author “Paul Baum” was partially supported by NSF grant DMS-1200475.
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Aubert, AM., Baum, P., Plymen, R., Solleveld, M. (2016). Depth and the Local Langlands Correspondence. In: Ballmann, W., Blohmann, C., Faltings, G., Teichner, P., Zagier, D. (eds) Arbeitstagung Bonn 2013. Progress in Mathematics, vol 319. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43648-7_2
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