Generic irreducibility of parabolic induction for real reductive groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916319513804800 |
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| author | Renard, David |
| author_facet | Renard, David |
| contents | Let $G$ be a real reductive linear group in the Harish-Chandra class. Suppose that $P$ is a parabolic subgroup of $G$ with Langlands decomposition $P=MAN$. Let $π$ be an irreducible representation of the Levi factor $L=MA$. We give sufficient conditions on the infinitesimal character of $π$ for the induced representation $i_P^G(π)$ to be irreducible. In particular, we prove that if $π_M$ is an irreducible representation of $M$, then for a generic character $χ_ν$ of $A$, the induced representation $i_P^G(π_M\boxtimes χ_ν)$ is irreducible. Here the parameter $ν$ is in $\mathfrak{a}^*=(\mathrm{Lie}(A)\otimes_\mathbb R \mathbb C)^*$ and generic means outside a countable, locally finite union of hyperplanes which depends only on the infinitesimal character of $π$. Notice that there is no other assumption on $π$ or $π_M$ than being irreducible, so the result is not limited to generalised principal series or standard representations, for which the result is already well known. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11202 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generic irreducibility of parabolic induction for real reductive groups Renard, David Representation Theory 22E45 Let $G$ be a real reductive linear group in the Harish-Chandra class. Suppose that $P$ is a parabolic subgroup of $G$ with Langlands decomposition $P=MAN$. Let $π$ be an irreducible representation of the Levi factor $L=MA$. We give sufficient conditions on the infinitesimal character of $π$ for the induced representation $i_P^G(π)$ to be irreducible. In particular, we prove that if $π_M$ is an irreducible representation of $M$, then for a generic character $χ_ν$ of $A$, the induced representation $i_P^G(π_M\boxtimes χ_ν)$ is irreducible. Here the parameter $ν$ is in $\mathfrak{a}^*=(\mathrm{Lie}(A)\otimes_\mathbb R \mathbb C)^*$ and generic means outside a countable, locally finite union of hyperplanes which depends only on the infinitesimal character of $π$. Notice that there is no other assumption on $π$ or $π_M$ than being irreducible, so the result is not limited to generalised principal series or standard representations, for which the result is already well known. |
| title | Generic irreducibility of parabolic induction for real reductive groups |
| topic | Representation Theory 22E45 |
| url | https://arxiv.org/abs/2310.11202 |