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arXiv · 2406.17514

Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality

Zhicheng Wang
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Abstract

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}.

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Zhicheng Wang. 2024-06-25. Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality. https://arxiv.org/abs/2406.17514

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