Unitarity of highest weight Harish-Chandra modules and smoothness of Schubert varieties
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909950663458816 |
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| author | Bai, Zhanqiang Erickson, William Q. Hunziker, Markus Jiang, Jing |
| author_facet | Bai, Zhanqiang Erickson, William Q. Hunziker, Markus Jiang, Jing |
| contents | Let $G_{\mathbb{R}}$ be a Lie group of Hermitian type, and $L(λ)$ a highest weight Harish-Chandra module of $G_{\mathbb{R}}$ with highest weight $λ$. In this article, we exhibit a bijection between the set of connected Dynkin subdiagrams containing the noncompact simple root and the set of unitary highest weight modules $L(-wρ-ρ)$, where $ρ$ is half the sum of positive roots. We find that $L(-wρ-ρ)$ is unitary if and only if the Schubert variety $X(w)$ is smooth. We also give the cardinality of the set of unitary highest weight modules $L(-wρ-ρ)$ for each Kazhdan-Lusztig right cell. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08199 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unitarity of highest weight Harish-Chandra modules and smoothness of Schubert varieties Bai, Zhanqiang Erickson, William Q. Hunziker, Markus Jiang, Jing Representation Theory Combinatorics 22E47, 17B10 Let $G_{\mathbb{R}}$ be a Lie group of Hermitian type, and $L(λ)$ a highest weight Harish-Chandra module of $G_{\mathbb{R}}$ with highest weight $λ$. In this article, we exhibit a bijection between the set of connected Dynkin subdiagrams containing the noncompact simple root and the set of unitary highest weight modules $L(-wρ-ρ)$, where $ρ$ is half the sum of positive roots. We find that $L(-wρ-ρ)$ is unitary if and only if the Schubert variety $X(w)$ is smooth. We also give the cardinality of the set of unitary highest weight modules $L(-wρ-ρ)$ for each Kazhdan-Lusztig right cell. |
| title | Unitarity of highest weight Harish-Chandra modules and smoothness of Schubert varieties |
| topic | Representation Theory Combinatorics 22E47, 17B10 |
| url | https://arxiv.org/abs/2512.08199 |