Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 439–490 | Cite as

Ramified automorphic induction and zero-dimensional affine Deligne–Lusztig varieties

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Abstract

To any connected reductive group G over a non-archimedean local field F and to any maximal torus T of G, we attach a family of extended affine Deligne–Lusztig varieties (and families of torsors over them) over the residue field of F. This construction generalizes affine Deligne–Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G. For \(G = {{\mathrm{GL}}}_2\) in the equal characteristic case, we prove that our construction indeed realizes the automorphic induction from at most tamely ramified tori. Moreover, if the torus is purely tamely ramified, then the varieties realizing this correspondence turn out to be (quite complicate) combinatorial objects: they are zero-dimensional and reduced, i.e., just disjoint unions of points.

Keywords

Affine Deligne–Lusztig variety Automorphic induction Local Langlands correspondence Supercuspidal representations 

Mathematics Subject Classification

11S37 14M15 11F70 

Notes

Acknowledgements

The author is very grateful to Paul Hamacher, Christian Liedtke, Stephan Neupert, Peter Scholze and Eva Viehmann for helpful discussions concerning this work. He is especially grateful to Eva Viehmann for valuable comments concerning a preliminary version of this manuscript. The author was partially supported by European Research Council starting Grant 277889 ”Moduli spaces of local G-shtukas”.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Fakultät für Mathematik der Technischen Universität München - M11GarchingGermany
  2. 2.Institut de Mathématiques de JussieuParis cedex 05France

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