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

Ramified automorphic induction and zero-dimensional affine Deligne–Lusztig varieties 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 $$G = {{\mathrm{GL}}}_2$$ G = 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

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

, Volume 288 (2) – May 16, 2017
52 pages

/lp/springer_journal/ramified-automorphic-induction-and-zero-dimensional-affine-deligne-eIBnfY3ute
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1896-x
Publisher site
See Article on Publisher Site

### 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 $$G = {{\mathrm{GL}}}_2$$ G = 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.

### Journal

Mathematische ZeitschriftSpringer Journals

Published: May 16, 2017

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