Hopf algebra and the duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917747437338624 |
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| author | Chen, Zhe |
| author_facet | Chen, Zhe |
| contents | In this paper we study the space $C(\mathfrak{gl}_n(\mathbb{F}_q))$ of complex invariant functions on $\mathfrak{gl}_n(\mathbb{F}_q)$, through a Hopf algebra viewpoint. First, we consider a variant notion of Zelevinsky's PSH algebra defined over the real numbers $\mathbb{R}$. In particular, we show that two specific $\mathbb{R}$-lattices inside the complex Hopf algebra $\bigoplus_nC(\mathfrak{gl}_n(\mathbb{F}_q))$ are real PSH algebras, and that they do not descend to $\mathbb{Z}$. Then, among consequences, we prove that every element in $C(\mathfrak{gl}_n(\mathbb{F}_q))$ is a linear combination of Harish-Chandra inductions of Kawanaka's pre-cuspidal functions, and give a conceptual characterisation of duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$, which in turn allows us to give a new proof of a classical result of Kawanaka. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06191 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hopf algebra and the duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$ Chen, Zhe Representation Theory Quantum Algebra Rings and Algebras In this paper we study the space $C(\mathfrak{gl}_n(\mathbb{F}_q))$ of complex invariant functions on $\mathfrak{gl}_n(\mathbb{F}_q)$, through a Hopf algebra viewpoint. First, we consider a variant notion of Zelevinsky's PSH algebra defined over the real numbers $\mathbb{R}$. In particular, we show that two specific $\mathbb{R}$-lattices inside the complex Hopf algebra $\bigoplus_nC(\mathfrak{gl}_n(\mathbb{F}_q))$ are real PSH algebras, and that they do not descend to $\mathbb{Z}$. Then, among consequences, we prove that every element in $C(\mathfrak{gl}_n(\mathbb{F}_q))$ is a linear combination of Harish-Chandra inductions of Kawanaka's pre-cuspidal functions, and give a conceptual characterisation of duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$, which in turn allows us to give a new proof of a classical result of Kawanaka. |
| title | Hopf algebra and the duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$ |
| topic | Representation Theory Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2408.06191 |