Représentations des quaternions de norme 1
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913700351311872 |
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| author | Henniart, Guy Vignéras, Marie-France |
| author_facet | Henniart, Guy Vignéras, Marie-France |
| contents | Let F be a local field with finite residue field of characteristic p, D the quaternion division algebra with centre F, and R an algebraically closed field of any characteristic. We classify the smooth irreducible R-representations V of the group D^1 of elements of D* with reduced norm 1. Such a V occurs in the restriction of a smooth irreducible R-representation V* of D*. When the dimension of V*is >1, following our previous work in the case of SL_2(F), we show that the restriction of V* to D^1 is irreducible or the sum of two irreducible representations. When the characteristic of R is not p, that restriction is the sum of two irreducible equivalent representations if and only if the representation of GL_2(F) image of V* by the Jacquet-Langlands correspondence restricts to SL_2(F) as a sum of four inequivalent irreducible representations (this is never the case if the characteristic of R is not 2). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Représentations des quaternions de norme 1 Henniart, Guy Vignéras, Marie-France Representation Theory 22E50, 11F70 Let F be a local field with finite residue field of characteristic p, D the quaternion division algebra with centre F, and R an algebraically closed field of any characteristic. We classify the smooth irreducible R-representations V of the group D^1 of elements of D* with reduced norm 1. Such a V occurs in the restriction of a smooth irreducible R-representation V* of D*. When the dimension of V*is >1, following our previous work in the case of SL_2(F), we show that the restriction of V* to D^1 is irreducible or the sum of two irreducible representations. When the characteristic of R is not p, that restriction is the sum of two irreducible equivalent representations if and only if the representation of GL_2(F) image of V* by the Jacquet-Langlands correspondence restricts to SL_2(F) as a sum of four inequivalent irreducible representations (this is never the case if the characteristic of R is not 2). |
| title | Représentations des quaternions de norme 1 |
| topic | Representation Theory 22E50, 11F70 |
| url | https://arxiv.org/abs/2502.14346 |