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On the Irreducibility of p-Adic Banach Principal Series of p-Adic \(\textrm{GL}_3\)

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Abstract

We establish an optimal (topological) irreducibility criterion for p-adic Banach principal series of \(\textrm{GL}_{n}(F)\), where \(F/\mathbb {Q}_p\) is finite and \(n \le 3\). This is new for \(n = 3\) as well as for \(n = 2\), \(F \ne \mathbb {Q}_p\) and establishes a refined version of Schneider’s conjecture (Schneider, P.: International congress of Mathematics, vol. II, pp. 1261–1282, 2006, Conjecture 2.5) for these groups.

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Acknowledgements

The first-named author was supported by JSPS KAKENHI Grant Number 18H01107. The second-named author was partially supported by an NSERC grant. We thank the referees for helpful comments. In particular, we thank the referee who asked rationality questions which led us to remove the assumption that the coefficient field C be sufficiently large. Part of this work was done during a pleasant stay of the first-named author at University of Toronto.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914, Tokyo, Japan

    Noriyuki Abe

  2. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Florian Herzig

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  1. Noriyuki Abe
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  2. Florian Herzig
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Correspondence to Florian Herzig.

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Dedicated to Pham Huu Tiep on the occasion of his 60th birthday.

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Abe, N., Herzig, F. On the Irreducibility of p-Adic Banach Principal Series of p-Adic \(\textrm{GL}_3\). Vietnam J. Math. 52, 451–478 (2024). https://doi.org/10.1007/s10013-023-00675-7

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  • DOI: https://doi.org/10.1007/s10013-023-00675-7

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