A fusion construction of local L-factors
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929347271589888 |
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| author | Bezrukavnikov, Roman Braverman, Alexander Finkelberg, Michael Kazhdan, David |
| author_facet | Bezrukavnikov, Roman Braverman, Alexander Finkelberg, Michael Kazhdan, David |
| contents | We propose a new conjectural way to calculate the local $L$-factor $L=L_χ(π,ρ,s)$ where $π$ is a representation of a $p$-adic group $G$, $ρ$ is an algebraic representation of the dual group $G^{\vee}$ and $χ$ is an algebraic character of $G$ satisfying a positivity condition.
A method going back to Godement and Jacquet yields a description of $L$ using as an input a certain space ${\mathcal S}_ρ$ of functions on $G$ depending on $ρ$. A (partly conjectural) description of ${\mathcal S}_ρ$ involving trace of Frobenius functions associated to perverse sheaves on the loop space of a semigroup containing $G$ was developed %by Bouthier, Ngo and Sakellaridis, partly based on an earlier work of Braverman and Kazhdan. Here we propose a different, more general conjectural description of ${\mathcal S}_ρ$: it also refers to trace of Frobenius functions but instead of the loop space of a semi-group we work with the ramified global Grassmannian fibering over the configuration space of points on a global curve defined by Beilinson-Drinfeld and Gaitsgory (a relation between two approaches is discussed in the appendix).
Our main result asserts validity of our conjectures where $π$ is generated by an Iwahori fixed vector: we show that in this case it is compatible with the standard formula for $L$ involving local Langlands correspondence which is known for such representations $π$.
The proof is based on properties of the coherent realization of the affine Hecke category. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_00913 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A fusion construction of local L-factors Bezrukavnikov, Roman Braverman, Alexander Finkelberg, Michael Kazhdan, David Representation Theory Algebraic Geometry Number Theory We propose a new conjectural way to calculate the local $L$-factor $L=L_χ(π,ρ,s)$ where $π$ is a representation of a $p$-adic group $G$, $ρ$ is an algebraic representation of the dual group $G^{\vee}$ and $χ$ is an algebraic character of $G$ satisfying a positivity condition. A method going back to Godement and Jacquet yields a description of $L$ using as an input a certain space ${\mathcal S}_ρ$ of functions on $G$ depending on $ρ$. A (partly conjectural) description of ${\mathcal S}_ρ$ involving trace of Frobenius functions associated to perverse sheaves on the loop space of a semigroup containing $G$ was developed %by Bouthier, Ngo and Sakellaridis, partly based on an earlier work of Braverman and Kazhdan. Here we propose a different, more general conjectural description of ${\mathcal S}_ρ$: it also refers to trace of Frobenius functions but instead of the loop space of a semi-group we work with the ramified global Grassmannian fibering over the configuration space of points on a global curve defined by Beilinson-Drinfeld and Gaitsgory (a relation between two approaches is discussed in the appendix). Our main result asserts validity of our conjectures where $π$ is generated by an Iwahori fixed vector: we show that in this case it is compatible with the standard formula for $L$ involving local Langlands correspondence which is known for such representations $π$. The proof is based on properties of the coherent realization of the affine Hecke category. |
| title | A fusion construction of local L-factors |
| topic | Representation Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2303.00913 |