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Local descent to quasi-split even general spin groups

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Let \(n>1\) and let \(\tau \) be an irreducible unitary supercuspidal representation of \({\text {GL}}_{2n}\) over a local non-archimedean field. Assuming the twisted symmetric square L-function of \(\tau \) has a pole at \(s=0\), we construct the local descent of \(\tau \) to the corresponding quasi-split even general spin group \({\text {GSpin}}_{2n}\). We prove this local descent is generic, unitary, supercuspidal and multiplicity free. Its irreducible quotients are “functorially related” to \(\tau \), in the analytic sense of a pole of a Rankin–Selberg type \(\gamma \)-function.

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Acknowledgements

We would like to thank Mahdi Asgari, Mikhail Borovoi, Wee Teck Gan, Dmitry Gourevitch, Joseph Hundley, Eitan Sayag, Freydoon Shahidi, David Soudry and Lei Zhang, for useful correspondences and helpful conversations. We would also like to thank the referees for many helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, 5290002, Ramat Gan, Israel

    Eyal Kaplan

  2. Singapore University of Social Sciences, 463 Clementi Road, Singapore, 599494, Singapore

    Jing Feng Lau

  3. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907-2067, USA

    Baiying Liu

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  1. Eyal Kaplan
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Correspondence to Baiying Liu.

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This research was supported by the ISRAEL SCIENCE FOUNDATION Grant nos. 376/21 and 421/17 (Kaplan), and by NSF Grants DMS-1702218, DMS-1848058, and start-up funds from the Department of Mathematics at Purdue University (Liu).

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Kaplan, E., Lau, J.F. & Liu, B. Local descent to quasi-split even general spin groups. Math. Z. 303, 69 (2023). https://doi.org/10.1007/s00209-023-03227-4

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