Equal rank local theta correspondence as a strong Morita equivalence
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916326578061312 |
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| author | Mesland, Bram Sengun, Mehmet Haluk |
| author_facet | Mesland, Bram Sengun, Mehmet Haluk |
| contents | Let (G,H) be one of the equal rank reductive dual pairs (Mp_{2n},O_{2n+1}) or (U_n,U_n) over a non-archimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say R(G) and R(H), 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 R(G) and R(H). 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13484 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Equal rank local theta correspondence as a strong Morita equivalence Mesland, Bram Sengun, Mehmet Haluk Representation Theory K-Theory and Homology Number Theory Operator Algebras 11F27, 22E50, 22D25, 46L80 Let (G,H) be one of the equal rank reductive dual pairs (Mp_{2n},O_{2n+1}) or (U_n,U_n) over a non-archimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say R(G) and R(H), 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 R(G) and R(H). 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. |
| title | Equal rank local theta correspondence as a strong Morita equivalence |
| topic | Representation Theory K-Theory and Homology Number Theory Operator Algebras 11F27, 22E50, 22D25, 46L80 |
| url | https://arxiv.org/abs/2207.13484 |