Pseudodifferential operators and the Connes-Kasparov isomorphism
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915870229397504 |
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| author | DeBello, Peter Higson, Nigel |
| author_facet | DeBello, Peter Higson, Nigel |
| contents | We compute the K-theory of the C*-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group G. Our result uses the Connes-Kasparov isomorphism for G, and in fact is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan's well-known parametrization of the tempered irreducible representations of G with real infinitesimal character. When the reductive group G has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a K-theoretic version of Vogan's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14985 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudodifferential operators and the Connes-Kasparov isomorphism DeBello, Peter Higson, Nigel K-Theory and Homology Operator Algebras Representation Theory 22E45, 22E46, 46L80 We compute the K-theory of the C*-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group G. Our result uses the Connes-Kasparov isomorphism for G, and in fact is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan's well-known parametrization of the tempered irreducible representations of G with real infinitesimal character. When the reductive group G has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a K-theoretic version of Vogan's theorem. |
| title | Pseudodifferential operators and the Connes-Kasparov isomorphism |
| topic | K-Theory and Homology Operator Algebras Representation Theory 22E45, 22E46, 46L80 |
| url | https://arxiv.org/abs/2502.14985 |