Local parameters of supercuspidal representations
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929596567388160 |
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| author | Gan, Wee Teck Harris, Michael Sawin, Will Beuzart-Plessis, Raphaël |
| author_facet | Gan, Wee Teck Harris, Michael Sawin, Will Beuzart-Plessis, Raphaël |
| contents | For a connected reductive group $G$ over a non-archime\-dean local field $F$ of positive characteristic, Genestier and Lafforgue have attached a semisimple parameter $\CL^{ss}(π)$ to each irreducible representation $π$. Our first result shows that the Genestier-Lafforgue parameter of a tempered $π$ can be uniquely refined to a tempered L-parameter $\CL(π)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(π)$ for unramfied $G$ and supercuspidal $π$ constructed by induction from an open compact (modulo center) subgroup. If $L^{ss}(π)$ is pure in an appropriate sense, we show that $\CL^{ss}(π)$ is ramified (unless $G$ is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal{L}^{ss}(π)$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_07737 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Local parameters of supercuspidal representations Gan, Wee Teck Harris, Michael Sawin, Will Beuzart-Plessis, Raphaël Representation Theory Number Theory 22E50, 11S37, 11F80, 11F70 For a connected reductive group $G$ over a non-archime\-dean local field $F$ of positive characteristic, Genestier and Lafforgue have attached a semisimple parameter $\CL^{ss}(π)$ to each irreducible representation $π$. Our first result shows that the Genestier-Lafforgue parameter of a tempered $π$ can be uniquely refined to a tempered L-parameter $\CL(π)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(π)$ for unramfied $G$ and supercuspidal $π$ constructed by induction from an open compact (modulo center) subgroup. If $L^{ss}(π)$ is pure in an appropriate sense, we show that $\CL^{ss}(π)$ is ramified (unless $G$ is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal{L}^{ss}(π)$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II. |
| title | Local parameters of supercuspidal representations |
| topic | Representation Theory Number Theory 22E50, 11S37, 11F80, 11F70 |
| url | https://arxiv.org/abs/2109.07737 |