On the Macdonald correspondence
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910773477900288 |
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| author | Aubert, Anne-Marie |
| author_facet | Aubert, Anne-Marie |
| contents | In 1980 Ian G. Macdonald established an explicit bijection between the isomorphism classes of the irreducible representations of ${\mathrm{GL}}_n(k)$, where $k$ is a finite field, and inertia equivalence classes of admissible tamely ramified $n$-dimensional Weil-Deligne representations of $W_F$, where $F$ is a non-archimedean local field with residue field $k$ and $W_F$ the absolute Weil group of $F$.
We describe a construction of the Macdonald correspondence based on the specialization to ${\mathrm{GL}}_n(k)$ of Lusztig's classification of irreducible representations of finite groups of Lie type, and review some properties of the correspondence. We define $ε$-factors for pairs of irreducible cuspidal representations of finite general linear groups, and show that they match with the expected Deligne $ε$-factors under the Macdonald correspondence. We use these $ε$-factors for pairs to obtain a characterization of the Macdonald correspondence for the irreducible cuspidal representations |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Macdonald correspondence Aubert, Anne-Marie Representation Theory 20C015 In 1980 Ian G. Macdonald established an explicit bijection between the isomorphism classes of the irreducible representations of ${\mathrm{GL}}_n(k)$, where $k$ is a finite field, and inertia equivalence classes of admissible tamely ramified $n$-dimensional Weil-Deligne representations of $W_F$, where $F$ is a non-archimedean local field with residue field $k$ and $W_F$ the absolute Weil group of $F$. We describe a construction of the Macdonald correspondence based on the specialization to ${\mathrm{GL}}_n(k)$ of Lusztig's classification of irreducible representations of finite groups of Lie type, and review some properties of the correspondence. We define $ε$-factors for pairs of irreducible cuspidal representations of finite general linear groups, and show that they match with the expected Deligne $ε$-factors under the Macdonald correspondence. We use these $ε$-factors for pairs to obtain a characterization of the Macdonald correspondence for the irreducible cuspidal representations |
| title | On the Macdonald correspondence |
| topic | Representation Theory 20C015 |
| url | https://arxiv.org/abs/2501.02332 |