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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.

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Notes

  1. The more canonical choice of all Frobenius lifts \(\varSigma ^{\prime } := \{ \sigma , \sigma \tau \}\) was suggested to the author by P. Scholze. At least in the cases we study in this article, this choice will lead to the same results as \(\varSigma \) from the text.

  2. A subtlety: we suppressed our choice of an identification of \(E^{\times }\) with the diagonal quotient of \(\tilde{I}_{m,w}\), for which we silently have chosen that u corresponds to \(y_1\). This choice determines on the one hand that \(y_1\) acts in \(V_{\chi }\) by \(\chi (u)\), and on the other hand, that we have to evaluate the trace formula using the identifications \(\varpi \leftrightarrow u \leftrightarrow y_1 = e_0(u,-u)\).

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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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  1. Alexander B. Ivanov

    Present address: Institut de Mathématiques de Jussieu, 4, place Jussieu, 75252, Paris cedex 05, France

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  1. Fakultät für Mathematik der Technischen Universität München - M11, Boltzmannstr. 3, 85748, Garching, Germany

    Alexander B. Ivanov

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Ivanov, A.B. Ramified automorphic induction and zero-dimensional affine Deligne–Lusztig varieties. Math. Z. 288, 439–490 (2018). https://doi.org/10.1007/s00209-017-1896-x

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