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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2304.06622 |
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| _version_ | 1866918468528373760 |
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| author | Deshpande, Tanmay Wagh, Saniya |
| author_facet | Deshpande, Tanmay Wagh, Saniya |
| contents | Let $\breve{K}$ be a complete discrete valuation field with an algebraically closed residue field ${k}$ and ring of integers $\breve{O}$. Let $T$ be a torus defined over $\breve{K}$. Let $L^+T$ denote the connected commutative pro-algebraic group over ${k}$ obtained by applying the Greenberg functor to the connected Néron model of $T$ over $\breve{O}$. Following the work of Serre for the multiplicative group, we first compute the fundamental group $π_1(L^+T)$. We then study multiplicative local systems (or character sheaves) on $L^+T$ and establish a local Langlands correspondence for them. Namely, we construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on $L^+T$ and inertial local Langlands parameters for $T$. Finally, we relate our results to the classical local Langlands correspondence for tori over local fields due to Langlands, via the sheaf-function correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_06622 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Character Sheaves on Tori over Local Fields Deshpande, Tanmay Wagh, Saniya Representation Theory Number Theory 20C Let $\breve{K}$ be a complete discrete valuation field with an algebraically closed residue field ${k}$ and ring of integers $\breve{O}$. Let $T$ be a torus defined over $\breve{K}$. Let $L^+T$ denote the connected commutative pro-algebraic group over ${k}$ obtained by applying the Greenberg functor to the connected Néron model of $T$ over $\breve{O}$. Following the work of Serre for the multiplicative group, we first compute the fundamental group $π_1(L^+T)$. We then study multiplicative local systems (or character sheaves) on $L^+T$ and establish a local Langlands correspondence for them. Namely, we construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on $L^+T$ and inertial local Langlands parameters for $T$. Finally, we relate our results to the classical local Langlands correspondence for tori over local fields due to Langlands, via the sheaf-function correspondence. |
| title | Character Sheaves on Tori over Local Fields |
| topic | Representation Theory Number Theory 20C |
| url | https://arxiv.org/abs/2304.06622 |