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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2306.12938 |
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| _version_ | 1866913981047767040 |
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| author | Mondal, Amiya Kumar Pattanayak, Basudev |
| author_facet | Mondal, Amiya Kumar Pattanayak, Basudev |
| contents | The main question we are going to address in this paper is: How much does the representation theory of the $p$-adic group $\mathrm{GL}_n(\mathcal{D})$ depend on the $p$-adic division algebra $\mathcal{D}$? Let $\mathcal{D}$ be a central division algebra defined over some locally compact non-archimedean local field. Using Bushnell-Kutzko theory of types and Sécherre-Stevens decomposition of spherical Hecke algebras associated to types, we obtain that the cuspidal blocks in the Bernstein decomposition of the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ of smooth complex representations of $\mathrm{GL}_n(\mathcal{D})$ do not depend on the $p$-adic division algebra $\mathcal{D}$. In particular, when $n=1$ or $2$, the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ does not depend on the $p$-adic division algebra $\mathcal{D}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12938 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smooth representations and Hecke algebras of $p$-adic $\mathrm{GL}_n(\mathcal{D})$ Mondal, Amiya Kumar Pattanayak, Basudev Representation Theory The main question we are going to address in this paper is: How much does the representation theory of the $p$-adic group $\mathrm{GL}_n(\mathcal{D})$ depend on the $p$-adic division algebra $\mathcal{D}$? Let $\mathcal{D}$ be a central division algebra defined over some locally compact non-archimedean local field. Using Bushnell-Kutzko theory of types and Sécherre-Stevens decomposition of spherical Hecke algebras associated to types, we obtain that the cuspidal blocks in the Bernstein decomposition of the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ of smooth complex representations of $\mathrm{GL}_n(\mathcal{D})$ do not depend on the $p$-adic division algebra $\mathcal{D}$. In particular, when $n=1$ or $2$, the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ does not depend on the $p$-adic division algebra $\mathcal{D}$. |
| title | Smooth representations and Hecke algebras of $p$-adic $\mathrm{GL}_n(\mathcal{D})$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2306.12938 |