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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.11696 |
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| _version_ | 1866911316074037248 |
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| author | Pattanayak, Basudev |
| author_facet | Pattanayak, Basudev |
| contents | This paper proves the branching laws for the full class of unitarizable representations of general linear groups in non-Archimedean local fields, extending the original notion of Gan-Gross-Prasad relevant pair for Arthur-type representations \cite{GGP2, Gur, Cha_crelle}. Further, we provide an explicit computable algorithm to determine the generalized GGP relevant pair, as developed in \cite{Cha_qbl}. In particular, we show that if $π$ and $π'$ are any irreducible smooth representations of $\mathrm{GL_{n+1}(F)}$ and $\mathrm{GL_{n}(F)}$ respectively, and their Langlands data or Zelevinsky data are given in terms of multisegments, then through an algorithmic process we can determine whether the space $\mathrm{Hom}_{\mathrm{GL_n(F)}}(π, π')$ is non-zero. Finally, when one of the represntations $π$ and $ π'$ is a generalized Speh representation, we give a complete classification for the other one for which the Hom space is non-zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quotient branching laws and Gan-Gross-Prasad relevance for general linear groups Pattanayak, Basudev Representation Theory This paper proves the branching laws for the full class of unitarizable representations of general linear groups in non-Archimedean local fields, extending the original notion of Gan-Gross-Prasad relevant pair for Arthur-type representations \cite{GGP2, Gur, Cha_crelle}. Further, we provide an explicit computable algorithm to determine the generalized GGP relevant pair, as developed in \cite{Cha_qbl}. In particular, we show that if $π$ and $π'$ are any irreducible smooth representations of $\mathrm{GL_{n+1}(F)}$ and $\mathrm{GL_{n}(F)}$ respectively, and their Langlands data or Zelevinsky data are given in terms of multisegments, then through an algorithmic process we can determine whether the space $\mathrm{Hom}_{\mathrm{GL_n(F)}}(π, π')$ is non-zero. Finally, when one of the represntations $π$ and $ π'$ is a generalized Speh representation, we give a complete classification for the other one for which the Hom space is non-zero. |
| title | Quotient branching laws and Gan-Gross-Prasad relevance for general linear groups |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2512.11696 |