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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2412.11604 |
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| _version_ | 1866915065767133184 |
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| author | Gerasimov, A. A. Lebedev, D. R. Oblezin, S. V. |
| author_facet | Gerasimov, A. A. Lebedev, D. R. Oblezin, S. V. |
| contents | The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator was introduced as an element of the $O_{\ell+1}$-spherical Hecke algebra associated with the Gelfand pair $O_{\ell+1}\subset GL_{\ell+1}(\mathbb{R})$. It was specified by the property to act on an $O_{\ell+1}$-fixed vector in a $GL_{\ell+1}(\mathbb{R})$-principal series representation via multiplication by the local Archimedean $L$-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group $GL_{\ell+1}(\mathbb{R})\times GL_{\ell+1}(\mathbb{R})$ by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group $Sp_{2\ell+2}(\mathbb{R})\times Sp_{2\ell+2}(\mathbb{R})$ by a Heisenberg Lie group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hecke-Baxter operators via Heisenberg group extensions Gerasimov, A. A. Lebedev, D. R. Oblezin, S. V. Representation Theory The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator was introduced as an element of the $O_{\ell+1}$-spherical Hecke algebra associated with the Gelfand pair $O_{\ell+1}\subset GL_{\ell+1}(\mathbb{R})$. It was specified by the property to act on an $O_{\ell+1}$-fixed vector in a $GL_{\ell+1}(\mathbb{R})$-principal series representation via multiplication by the local Archimedean $L$-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group $GL_{\ell+1}(\mathbb{R})\times GL_{\ell+1}(\mathbb{R})$ by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group $Sp_{2\ell+2}(\mathbb{R})\times Sp_{2\ell+2}(\mathbb{R})$ by a Heisenberg Lie group. |
| title | The Hecke-Baxter operators via Heisenberg group extensions |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2412.11604 |