Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917260693602304 |
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| author | Sorgdrager, Reinier |
| author_facet | Sorgdrager, Reinier |
| contents | We prove that an admissible $p$-adic Banach representation of $\text{GL}_2K$ whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension $\leq[K\colon\mathbf Q_p]$, where $p>2$ and $K$ is a $p$-adic field. We also prove this for the group of units of the quaternions over $K$ replacing $\text{GL}_2K$. In the process, we make some observations in the theory of $p$-adic Banach representations that might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08856 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units Sorgdrager, Reinier Representation Theory Number Theory We prove that an admissible $p$-adic Banach representation of $\text{GL}_2K$ whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension $\leq[K\colon\mathbf Q_p]$, where $p>2$ and $K$ is a $p$-adic field. We also prove this for the group of units of the quaternions over $K$ replacing $\text{GL}_2K$. In the process, we make some observations in the theory of $p$-adic Banach representations that might be of independent interest. |
| title | Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2602.08856 |