Irreducibility of the parabolic induction of essentially Speh representations and a representation of Arthur type over a p-adic field
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909883733901312 |
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| author | Bošnjak, Barbara Stadler, Alexander |
| author_facet | Bošnjak, Barbara Stadler, Alexander |
| contents | Let $F$ be a $p$-adic field. In this article, we consider representations of split special orthogonal groups $\mathrm{SO}_{2n+1}(F)$ and symplectic groups $\mathrm{Sp}_{2n}(F)$ of rank $n$. We denote by $π_1 \times \ldots \times π_r \rtimes π$ the normalized parabolically induced representation of either. Now let $u_i$ be essentially Speh representations and $π$ a representation of Arthur type. We prove that the parabolic induction $u_1 \times \ldots \times u_r \rtimes π$ is irreducible if and only if $u_i \times u_j$, $u_i \times u_j^\vee$ and $u_i \rtimes π$ are irreducible for all choices of $i\neq j$. If $u_i$ are Speh representations, we determine the composition series of the above parabolically induced representation. Through this, we are able to produce a new collection of unitary representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irreducibility of the parabolic induction of essentially Speh representations and a representation of Arthur type over a p-adic field Bošnjak, Barbara Stadler, Alexander Representation Theory Let $F$ be a $p$-adic field. In this article, we consider representations of split special orthogonal groups $\mathrm{SO}_{2n+1}(F)$ and symplectic groups $\mathrm{Sp}_{2n}(F)$ of rank $n$. We denote by $π_1 \times \ldots \times π_r \rtimes π$ the normalized parabolically induced representation of either. Now let $u_i$ be essentially Speh representations and $π$ a representation of Arthur type. We prove that the parabolic induction $u_1 \times \ldots \times u_r \rtimes π$ is irreducible if and only if $u_i \times u_j$, $u_i \times u_j^\vee$ and $u_i \rtimes π$ are irreducible for all choices of $i\neq j$. If $u_i$ are Speh representations, we determine the composition series of the above parabolically induced representation. Through this, we are able to produce a new collection of unitary representations. |
| title | Irreducibility of the parabolic induction of essentially Speh representations and a representation of Arthur type over a p-adic field |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2504.01226 |