Abstract
The affinoid enveloping algebra \(\widehat {U({\mathscr{L}})}_{K}\) of a free, finitely generated \(\mathbb {Z}_{p}\)-Lie algebra \({\mathscr{L}}\) has proven to be useful within the representation theory of compact p-adic Lie groups, and we aim to further understand its algebraic structure. To this end, we define the notion of a Dixmier module over \(\widehat {U({\mathscr{L}})}_{K}\), a generalisation of the Verma module, and we prove that when \({\mathscr{L}}\) is nilpotent, all primitive ideals of \(\widehat {U({\mathscr{L}})}_{K}\) can be described in terms of annihilator ideals of Dixmier modules. Using this, we take steps towards proving that this algebra satisfies a version of the classical Dixmier-Moeglin equivalence.
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Acknowledgements
I am very grateful to Konstantin Ardakov and Ioan Stanciu for many helpful conversations and discussions. I would also like to thank EPSRC and the Heilbronn Institute for Mathematical Research for supporting and funding this research.
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Presented by: Anne Moreau
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Jones, A. Affinoid Dixmier Modules and the Deformed Dixmier-Moeglin Equivalence. Algebr Represent Theor 26, 23–70 (2023). https://doi.org/10.1007/s10468-021-10084-4
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DOI: https://doi.org/10.1007/s10468-021-10084-4