On a geometric comparison of representations of complex and $p$-adic $\mathbf{GL}_n$
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912971521785856 |
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| author | Deng, Taiwang Huang, Chang Xu, Bin Zhao, Qixian |
| author_facet | Deng, Taiwang Huang, Chang Xu, Bin Zhao, Qixian |
| contents | In this paper, we use geometric methods to study the relations between admissible representations of $\mathbf{GL}_n(\mathbb{C})$ and unramified representations of $\mathbf{GL}_m(\mathbb{Q}_p)$. We show that the geometric relationship between Langlands parameter spaces of $\mathbf{GL}_n(\mathbb{C})$ and $\mathbf{GL}_m(\mathbb{Q}_p)$ constructed by the first named author is compatible with the functor recently defined algebraically by Chan-Wong. We then show that the said relationship intertwines translation functors on representations of $\mathbf{GL}_n(\mathbb{C})$ and partial Bernstein-Zelevinskii derivatives on representations of $\mathbf{GL}_m(\mathbb{Q}_p)$, providing purely geometric counterparts to some results of Chan-Wong. In the sequels, the techniques of this work will be extended to real and $p$-adic classical groups and used to study their Arthur packets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15653 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a geometric comparison of representations of complex and $p$-adic $\mathbf{GL}_n$ Deng, Taiwang Huang, Chang Xu, Bin Zhao, Qixian Representation Theory 22E50 In this paper, we use geometric methods to study the relations between admissible representations of $\mathbf{GL}_n(\mathbb{C})$ and unramified representations of $\mathbf{GL}_m(\mathbb{Q}_p)$. We show that the geometric relationship between Langlands parameter spaces of $\mathbf{GL}_n(\mathbb{C})$ and $\mathbf{GL}_m(\mathbb{Q}_p)$ constructed by the first named author is compatible with the functor recently defined algebraically by Chan-Wong. We then show that the said relationship intertwines translation functors on representations of $\mathbf{GL}_n(\mathbb{C})$ and partial Bernstein-Zelevinskii derivatives on representations of $\mathbf{GL}_m(\mathbb{Q}_p)$, providing purely geometric counterparts to some results of Chan-Wong. In the sequels, the techniques of this work will be extended to real and $p$-adic classical groups and used to study their Arthur packets. |
| title | On a geometric comparison of representations of complex and $p$-adic $\mathbf{GL}_n$ |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2504.15653 |