On Gelfand pairs and degenerate Gelfand-Graev modules of General Linear groups of degree two over principal ideal local rings of finite length
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| Format: | Preprint |
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2024
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| _version_ | 1866913566975590400 |
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| author | Gupta, Archita Singla, Pooja |
| author_facet | Gupta, Archita Singla, Pooja |
| contents | Let $R$ be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Denote by $G(R)$ the general linear group of degree two over $R$, and by $B(R)$ the Borel subgroup of $G(R)$ consisting of upper triangular matrices. In this article, we prove that the pair $(G(R), B(R))$ is a strong Gelfand pair. We also investigate the decomposition of the degenerate Gelfand-Graev (DGG) modules of $G(R)$. It is known that the non-degenerate Gelfand Graev module (also called non-degenerate Whittaker model) of $G(R)$ is multiplicity-free. We characterize the DGG-modules where the multiplicities are independent of the cardinality of the residue field. We provide a complete decomposition of all DGG modules of $G(R)$ for $R$ of length at most four. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_21761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Gelfand pairs and degenerate Gelfand-Graev modules of General Linear groups of degree two over principal ideal local rings of finite length Gupta, Archita Singla, Pooja Representation Theory Primary 20G05, Secondary 20C15, 20G25, 15B33 Let $R$ be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Denote by $G(R)$ the general linear group of degree two over $R$, and by $B(R)$ the Borel subgroup of $G(R)$ consisting of upper triangular matrices. In this article, we prove that the pair $(G(R), B(R))$ is a strong Gelfand pair. We also investigate the decomposition of the degenerate Gelfand-Graev (DGG) modules of $G(R)$. It is known that the non-degenerate Gelfand Graev module (also called non-degenerate Whittaker model) of $G(R)$ is multiplicity-free. We characterize the DGG-modules where the multiplicities are independent of the cardinality of the residue field. We provide a complete decomposition of all DGG modules of $G(R)$ for $R$ of length at most four. |
| title | On Gelfand pairs and degenerate Gelfand-Graev modules of General Linear groups of degree two over principal ideal local rings of finite length |
| topic | Representation Theory Primary 20G05, Secondary 20C15, 20G25, 15B33 |
| url | https://arxiv.org/abs/2410.21761 |