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Cuspidal representations in the cohomology of Deligne–Lusztig varieties for GL(2) over finite rings

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Abstract

We define closed subvarieties of some Deligne–Lusztig varieties for GL(2) over finite rings and study their ´etale cohomology. As a result, we show that cuspidal representations appear in it. Such closed varieties are studied in [Lus2] in a special case. We can do the same things for a Deligne–Lusztig variety associated to a quaternion division algebra over a non-archimedean local field. A product of such varieties can be regarded as an affine bundle over a curve. The base curve appears as an open subscheme of a union of irreducible components of the stable reduction of the Lubin–Tate curve in a special case. Finally, we state some conjecture on a part of the stable reduction using the above varieties. This is an attempt to understand bad reduction of Lubin–Tate curves via Deligne–Lusztig varieties.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan

    Tetsushi Ito

  2. Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33 Yayoi-cho, Inage, Chiba, 263-8522, Japan

    Takahiro Tsushima

Authors
  1. Tetsushi Ito
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  2. Takahiro Tsushima
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Correspondence to Tetsushi Ito.

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Ito, T., Tsushima, T. Cuspidal representations in the cohomology of Deligne–Lusztig varieties for GL(2) over finite rings. Isr. J. Math. 226, 877–926 (2018). https://doi.org/10.1007/s11856-018-1717-x

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  • DOI: https://doi.org/10.1007/s11856-018-1717-x

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