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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2401.09302 |
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| _version_ | 1866929213105242112 |
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| author | André, Carlos A. M. Dias, João |
| author_facet | André, Carlos A. M. Dias, João |
| contents | An algebra group over a field $F$ is a group of the form $G = 1+J$ where $J$ is a finite-dimensional nilpotent associative $F$-algebra. A theorem of M. Boyarchenko asserts that, in the case where $F$ is a non-archimedean local field, every irreducible smooth representation of $G$ is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of $G$. If $J$ is a nilpotent algebra endowed with an involution $σ:J\to J$, then $σ$ naturally defines a group automorphism of $G$, and we may consider the fixed point subgroup $C_{G}(σ)$. Assuming that $F$ has characteristic different from $2$, we extend Boyarchenko's result and show that every irreducible smooth representation of $C_{G}(σ)$ is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form $C_{H}(σ)$ where $H$ is an $σ$-invariant algebra subgroup of $G$. As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over $F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09302 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smooth representations of involutive algebra groups over non-archimedean local fields André, Carlos A. M. Dias, João Representation Theory 20G25, 22D12, 22D30 An algebra group over a field $F$ is a group of the form $G = 1+J$ where $J$ is a finite-dimensional nilpotent associative $F$-algebra. A theorem of M. Boyarchenko asserts that, in the case where $F$ is a non-archimedean local field, every irreducible smooth representation of $G$ is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of $G$. If $J$ is a nilpotent algebra endowed with an involution $σ:J\to J$, then $σ$ naturally defines a group automorphism of $G$, and we may consider the fixed point subgroup $C_{G}(σ)$. Assuming that $F$ has characteristic different from $2$, we extend Boyarchenko's result and show that every irreducible smooth representation of $C_{G}(σ)$ is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form $C_{H}(σ)$ where $H$ is an $σ$-invariant algebra subgroup of $G$. As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over $F$. |
| title | Smooth representations of involutive algebra groups over non-archimedean local fields |
| topic | Representation Theory 20G25, 22D12, 22D30 |
| url | https://arxiv.org/abs/2401.09302 |