Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality
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| Format: | Preprint |
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2024
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| _version_ | 1866911932126068736 |
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| author | Wang, Zhicheng |
| author_facet | Wang, Zhicheng |
| contents | Lusztig \cite{L5,L6} gave a parametrization for $\rm{Irr}(G^F)$, where $G$ is a reductive algebraic group defined over $\mathbb{F}_q$, with Frobenius map $F$. This parametrization is known as Lusztig's Jordan decomposition or Lusztig correspondence. However, there is not a canonical choice of Lusztig correspondence. In this paper, we consider classical groups. We pick a canonical choice of Lusztig correspondence which is compatible with parabolic induction and is compatible with theta correspondence. This result extends Pan's result in \cite{P3}. As an application, we give a refinement of the results of the finite Gan-Gross-Prasad problem in \cite{Wang1} and prove a duality between Theta correspondence and finite Gan-Gross-Prasad problem, which can be regarded as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh \cite{BZSV}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17514 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality Wang, Zhicheng Representation Theory Lusztig \cite{L5,L6} gave a parametrization for $\rm{Irr}(G^F)$, where $G$ is a reductive algebraic group defined over $\mathbb{F}_q$, with Frobenius map $F$. This parametrization is known as Lusztig's Jordan decomposition or Lusztig correspondence. However, there is not a canonical choice of Lusztig correspondence. In this paper, we consider classical groups. We pick a canonical choice of Lusztig correspondence which is compatible with parabolic induction and is compatible with theta correspondence. This result extends Pan's result in \cite{P3}. As an application, we give a refinement of the results of the finite Gan-Gross-Prasad problem in \cite{Wang1} and prove a duality between Theta correspondence and finite Gan-Gross-Prasad problem, which can be regarded as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh \cite{BZSV}. |
| title | Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2406.17514 |