Coordinate rings on symmetric spaces
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929242044891136 |
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| author | Bao, Huanchen Song, Jinfeng |
| author_facet | Bao, Huanchen Song, Jinfeng |
| contents | Let $G_k$ be a connected reductive group over an algebraically closed field $k$ of char $\neq 2$. Let $θ_k$ be an algebraic group involution of $G_k$ and denote the fixed point subgroup by $K_k$. We construct an integral model for the symmetric space $K_k \backslash G_k$ with a natural action of the Chevalley group scheme over integers. We show the coordinate ring $k[K_k \backslash G_k]$ admits a canonical basis, as well as a good filtration as a $G_k$-module. We also construct a canonical basis and an integral form for the space of $K_k$-biinvariant functions on $k[G_k]$. Our results rely on the construction of quantized coordinate algebras of symmetric spaces, using the theory of canonical bases on quantum symmetric pairs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08258 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coordinate rings on symmetric spaces Bao, Huanchen Song, Jinfeng Representation Theory Algebraic Geometry Quantum Algebra 20G05, 20G42, 17B37, 14M27 Let $G_k$ be a connected reductive group over an algebraically closed field $k$ of char $\neq 2$. Let $θ_k$ be an algebraic group involution of $G_k$ and denote the fixed point subgroup by $K_k$. We construct an integral model for the symmetric space $K_k \backslash G_k$ with a natural action of the Chevalley group scheme over integers. We show the coordinate ring $k[K_k \backslash G_k]$ admits a canonical basis, as well as a good filtration as a $G_k$-module. We also construct a canonical basis and an integral form for the space of $K_k$-biinvariant functions on $k[G_k]$. Our results rely on the construction of quantized coordinate algebras of symmetric spaces, using the theory of canonical bases on quantum symmetric pairs. |
| title | Coordinate rings on symmetric spaces |
| topic | Representation Theory Algebraic Geometry Quantum Algebra 20G05, 20G42, 17B37, 14M27 |
| url | https://arxiv.org/abs/2402.08258 |