Induction automorphe: représentations unitaires et spectre résiduel
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916989236150272 |
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| author | Fatou, Martin Lemaire, Bertrand |
| author_facet | Fatou, Martin Lemaire, Bertrand |
| contents | Let $E/F$ be a finite cyclic extension of local fields of characteristic zero, of degree $d$, and $κ$ be a character of $F^\times$ whose kernel is $\mathrm{N}_{E/F}(E^\times)$. For $m\in \mathbb{N}^*$, we prove that every irreducible unitary representation of $\mathrm{GL}_m(E)$ has a $κ$-lift to $\mathrm{GL}_{md}(F)$, given by a character identity as in Henniart-Herb [HH]. Let ${\bf E}/{\bf F}$ be a finite cyclic extension of number fields, of degree $d$, and $\mathfrak{K}$ be a character of $\mathbb{A}_{\bf F}^\times$ whose kernel is ${\bf F}^\times \mathrm{N}_{{\bf E}/{\bf F}}(\mathbb{A}_{\bf E}^\times)$. We prove that every automorphic discrete representation of $\mathrm{GL}_m(\mathbb{A}_{\bf E})$ has a (strong) $\mathfrak{K}$-lift to $\mathrm{GL}_{md}(\mathbb{A}_{\bf F})$, i.e. compatible with the local lifting maps. We describe the image and the fibres of these local and global lifting maps. Locally, we also treat the elliptic representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02775 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Induction automorphe: représentations unitaires et spectre résiduel Fatou, Martin Lemaire, Bertrand Representation Theory Number Theory 22E50, 22E55, 11F70, 11F72 Let $E/F$ be a finite cyclic extension of local fields of characteristic zero, of degree $d$, and $κ$ be a character of $F^\times$ whose kernel is $\mathrm{N}_{E/F}(E^\times)$. For $m\in \mathbb{N}^*$, we prove that every irreducible unitary representation of $\mathrm{GL}_m(E)$ has a $κ$-lift to $\mathrm{GL}_{md}(F)$, given by a character identity as in Henniart-Herb [HH]. Let ${\bf E}/{\bf F}$ be a finite cyclic extension of number fields, of degree $d$, and $\mathfrak{K}$ be a character of $\mathbb{A}_{\bf F}^\times$ whose kernel is ${\bf F}^\times \mathrm{N}_{{\bf E}/{\bf F}}(\mathbb{A}_{\bf E}^\times)$. We prove that every automorphic discrete representation of $\mathrm{GL}_m(\mathbb{A}_{\bf E})$ has a (strong) $\mathfrak{K}$-lift to $\mathrm{GL}_{md}(\mathbb{A}_{\bf F})$, i.e. compatible with the local lifting maps. We describe the image and the fibres of these local and global lifting maps. Locally, we also treat the elliptic representations. |
| title | Induction automorphe: représentations unitaires et spectre résiduel |
| topic | Representation Theory Number Theory 22E50, 22E55, 11F70, 11F72 |
| url | https://arxiv.org/abs/2505.02775 |