The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator: principal series representations
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| Format: | Preprint |
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2025
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| _version_ | 1866909709795065856 |
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| author | Gerasimov, Anton A. Lebedev, Dmitry R. Oblezin, Sergey V. |
| author_facet | Gerasimov, Anton A. Lebedev, Dmitry R. Oblezin, Sergey V. |
| contents | Previously introduced the $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator is a one-parameter family of elements in the commutative spherical Hecke algebra $\mathcal{H}(GL_{\ell+1}(\mathbb{R}),O_{\ell+1})$. Its action on spherical vectors in spherical principle series representations of $GL_{\ell+1}(\mathbb{R})$ is given by multiplication by the Archimedean $L$-factors associated to these representations. In this note we propose an extension of the construction to other (non-spherical) $GL_{\ell+1}(\mathbb{R})$ principle series representations providing a relevant generalization of the notions of spherical vector, commutative spherical Hecke algebra and the Hecke-Baxter operator to the general case. Action of the introduced Hecke-Baxter operator on the generalized spherical vectors is given by multiplication by the Archiemdean $L$-factor associated to the corresponding principle series representation of $GL_{\ell+1}(\mathbb{R})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_16708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator: principal series representations Gerasimov, Anton A. Lebedev, Dmitry R. Oblezin, Sergey V. Representation Theory Number Theory 22D30, 22E45, 43A90 Previously introduced the $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator is a one-parameter family of elements in the commutative spherical Hecke algebra $\mathcal{H}(GL_{\ell+1}(\mathbb{R}),O_{\ell+1})$. Its action on spherical vectors in spherical principle series representations of $GL_{\ell+1}(\mathbb{R})$ is given by multiplication by the Archimedean $L$-factors associated to these representations. In this note we propose an extension of the construction to other (non-spherical) $GL_{\ell+1}(\mathbb{R})$ principle series representations providing a relevant generalization of the notions of spherical vector, commutative spherical Hecke algebra and the Hecke-Baxter operator to the general case. Action of the introduced Hecke-Baxter operator on the generalized spherical vectors is given by multiplication by the Archiemdean $L$-factor associated to the corresponding principle series representation of $GL_{\ell+1}(\mathbb{R})$. |
| title | The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator: principal series representations |
| topic | Representation Theory Number Theory 22D30, 22E45, 43A90 |
| url | https://arxiv.org/abs/2506.16708 |