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Quaternion distinguished generic representations of GL2n

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Abstract

Let E/F be a quadratic extension of non-Archimedean local fields of characteristic 0. Let D be the unique quaternion division algebra over F and fix an embedding of E to D. Then, GLm(D) can be regarded as a subgroup of GL2m(E). Using the method of Matringe, we classify irreducible generic GLm(D)-distinguished representations of GL2m(E) in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of E, we prove a sufficient condition for a generic representation in the image of the unstable base change lift from the unitary group U2m to be GLm(D)-distinguished.

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Acknowledgements

I would like to thank Tamotsu Ikeda, Hiraku Atobe and Masao Oi for many helpful comments. I am also grateful to the anonymous referees for pointing out some inaccuracies in an earlier draft, especially equation (3) in Lemma 4.2 and numerous comments which have greatly improved the manuscript. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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  1. Faculty of Mathematics and Physics, Institute of Science and Engineering Kanazawa University, Kakumamachi, Ishikawa, 920-1192, Japan

    Miyu Suzuki

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  1. Miyu Suzuki
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Correspondence to Miyu Suzuki.

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Suzuki, M. Quaternion distinguished generic representations of GL2n. Isr. J. Math. 238, 871–899 (2020). https://doi.org/10.1007/s11856-020-2045-5

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  • DOI: https://doi.org/10.1007/s11856-020-2045-5

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