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Normal elements in the Iwasawa algebras of Chevalley groups

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For a prime \(p>2\), let G be a semi-simple, simply connected, split Chevalley group over \({\mathbb {Z}}_p\), G(1) be the first congruence kernel of G and \(\Omega _{G(1)}\) be the mod-p Iwasawa algebra defined over the finite field \({\mathbb {F}}_p\). Ardakov et al. (Adv Math 218: 865–901, 2008) have shown that if p is a “nice prime ” (\(p \ge 5\) and \(p \not \mid (n+1)\) if the Lie algebra of G(1) is of type \(A_n\)), then every non-zero normal element in \(\Omega _{G(1)}\) is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov et al. result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture. We also propose some potential topics regarding to the normal elements and ideals in the Iwasawa algebras of the pro-p Iwahori subgroups of general linear group \({\hbox {GL}}_n(\mathbb {Z}_p)\) and discuss how to extend our current techniques and methods to the case of the pro-p Iwahori subgroups of \({\hbox {GL}}_n(\mathbb {Z}_p)\).

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Acknowledgements

We have accumulated quite a debt of gratitude in writing this paper. First of all to Professors John Coates and Peter Schneider, we express our warmest gratitude for their encouragement which kept this work progress everyday. Most of all to Professors Konstantin Ardakov and James J. Zhang for many unreserved discussions and for constant inspiration. We would like to thank the anonymous referees for his/her tremendous job, who, apart from two very thorough reports which helped to correct a number of minor errors, lacunae and other inaccuracies (both mathematical and pedagogical), also taught us some theory of p-adic Lie groups and p-adic Lie algebras. In particular, he/she suggested that we should reorganize the original manuscript and break up the proof of the main Theorem into several separate parts. He/she paid special attention to the future potential work of case \(p=2\) (cf. Sect. 8.1). And last but not least, we are sincerely grateful to Professor Benjamin Howard for his efficient reviewing process. We are thankful for his kind considerations and his advices on how to modify our manuscript. The second author would like to thank PIMS-CNRS and the University of British Columbia for postdoctoral research grant. He is also thankful to Beijing Institute of Technology (BIT) for its gracious hospitality during a visit on August 2018 when this collaboration took place. The main idea of this work grew out of his visit to BIT.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, People’s Republic of China

    Dong Han

  2. Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancourer, BC, V6T 1Z2, Canada

    Jishnu Ray

  3. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Feng Wei

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  1. Dong Han
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Correspondence to Feng Wei.

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The work of the first author is supported by National Natural Science Foundation of China under Grant 11926415. This work is partially supported by the Foreign High-level Cultural and Educational Experts Project of the Beijing Institute of Technology.

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Han, D., Ray, J. & Wei, F. Normal elements in the Iwasawa algebras of Chevalley groups. manuscripta math. 165, 415–451 (2021). https://doi.org/10.1007/s00229-020-01214-1

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