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A Note on Integral Structures in Some Locally Algebraic Representations of GL2

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Abstract

In the p-adic local Langlands correspondence for GL2 (ℚp), the following theorem of Berger and Breuil has played an important role: the locally algebraic representations of GL2(ℚp) associated to crystabelline Galois representations admit a unique unitary completion. In this note, we give a new proof of the weaker statement that the locally algebraic representations admit at most one unitary completion and such a completion is automatically admissible. Our proof is purely representation theoretic, involving neither (ϕ, Γ)-module techniques nor global methods. When F is a finite extension of ℚp, we also get a simpler proof of a theorem of Vignéras for the existence of integral structures for (locally algebraic) special series and for (smooth) tamely ramified principal series.

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Acknowledgements

We would like to thank V. Paškūnas for his comments, especially for his encouragement to publish this note. We also thank the anonymous referee for the careful reading of the manuscript and pertinent comments.

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Authors and Affiliations

  1. Morningside Center of Mathematics, Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing, 100190, P. R. China

    Yong Quan Hu

  2. Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, 100049, P. R. China

    Yong Quan Hu

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  1. Yong Quan Hu
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Correspondence to Yong Quan Hu.

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Supported by National Natural Science Foundation of China (Grant No. 11688101), China’s Recruitement Program of Global Experts, National Center for Mathematics and Interdisciplinary Sciences and Hua Loo-Keng Center for Mathematical Sciences of Chinese Academy of Sciences

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Hu, Y.Q. A Note on Integral Structures in Some Locally Algebraic Representations of GL2. Acta. Math. Sin.-English Ser. 37, 59–72 (2021). https://doi.org/10.1007/s10114-020-8213-z

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  • DOI: https://doi.org/10.1007/s10114-020-8213-z

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