The Hecke-Baxter operators via Heisenberg group extensions
| Autor: | Gerasimov, A. A., Lebedev, D. R., Oblezin, S. V. |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 20 pages |
| Databáze: | arXiv |
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