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

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}.