A characterization of unitarity of some highest weight Harish-Chandra modules
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912045523271680 |
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| author | Bai, Zhanqiang Hunziker, Markus |
| author_facet | Bai, Zhanqiang Hunziker, Markus |
| contents | Let $L(λ)$ be a highest weight Harish-Chandra module with highest weight $λ$. When the associated variety of $L(λ)$ is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of $L(λ)$ can be determined by a simple condition on the value of $z = (λ+ ρ, β^{\vee})$, where $ρ$ is half the sum of positive roots and $β$ is the highest root. In the proof, certain distinguished antichains of positive noncompact roots play a key role.
By using these antichains, we are also able to provide a uniform formula for the Gelfand--Kirillov dimension of all highest weight Harish-Chandra modules, generalizing our previous result for the case of unitary highest weight Harish-Chandra modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16555 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of unitarity of some highest weight Harish-Chandra modules Bai, Zhanqiang Hunziker, Markus Representation Theory 22E47, 17B10 Let $L(λ)$ be a highest weight Harish-Chandra module with highest weight $λ$. When the associated variety of $L(λ)$ is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of $L(λ)$ can be determined by a simple condition on the value of $z = (λ+ ρ, β^{\vee})$, where $ρ$ is half the sum of positive roots and $β$ is the highest root. In the proof, certain distinguished antichains of positive noncompact roots play a key role. By using these antichains, we are also able to provide a uniform formula for the Gelfand--Kirillov dimension of all highest weight Harish-Chandra modules, generalizing our previous result for the case of unitary highest weight Harish-Chandra modules. |
| title | A characterization of unitarity of some highest weight Harish-Chandra modules |
| topic | Representation Theory 22E47, 17B10 |
| url | https://arxiv.org/abs/2409.16555 |