Abstract
In this paper, we prove that there is at most one correspondence between parahoric-spherical representations and semisimple local Langlands parameters which satisfies certain natural properties. Our proof of this uniqueness statement is very formal. In particular, the semisimple local Langlands parameters constructed in Fargues and Scholze (Geometrization of the local Langlands correspondence, 2021. arXiv:2102.13459) yield the unique candidate for such representations. As a corollary, we prove the conjecture (Haines in IMRN 2015(20):10367–10398, 2015) [Conjecture 13.1] which posits the compatibility of semisimple local Langlands parameters with parahoric Satake parameters.
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Acknowledgements
It is the author’s pleasure to thank his advisor Thomas J. Haines for giving him this project, for several useful discussions about it, and also for helpful comments on the preliminary version. The author would like to thank Siyan Daniel Li-Huerta for explaining some concepts related to Genestier-Lafforgue’s parameters. He also really appreciates the kind suggestions given by the very responsible reviewer and editor, who helped to make this paper much more understandable. The author’s research is partially supported by a Hauptman Summer Fellowship (2021) at University of Maryland, and he sincerely thank the donor Carol Fullerton for her kind support.
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Research of Q.L. is partially supported by a Hauptman Summer Fellowship (2021) at University of Maryland.
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Li, Q. Compatibility of semisimple local Langlands parameters with parahoric Satake parameters. manuscripta math. 172, 669–683 (2023). https://doi.org/10.1007/s00229-022-01454-3
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DOI: https://doi.org/10.1007/s00229-022-01454-3