Mesland, B. and Şengün, M.H. (2024) Equal rank local theta correspondence as a strong Morita equivalence. Selecta Mathematica, 30 (4). 72. ISSN 1022-1824
Abstract
Let (G, H) be one of the equal rank reductive dual pairs (Mp2n, O2n+1) or (Un, Un) over a nonarchimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say G θ and H θ , of the tempered duals of G and H. We prove that this bijection arises from an equivalence between the categories of representations of two C∗-algebras whose spectra are Gθ and Hθ . This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2024. Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Funding Information: | Funder Grant number ENGINEERING AND PHYSICAL SCIENCE RESEARCH COUNCIL EP/V049119/1 |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 01 Aug 2024 10:35 |
Last Modified: | 01 Aug 2024 10:35 |
Status: | Published |
Publisher: | Springer Science and Business Media LLC |
Refereed: | Yes |
Identification Number: | 10.1007/s00029-024-00966-y |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:215358 |