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

Lusztig {L5,L6} gave a parametrization for Irr(GF), where G is a reductive algebraic group defined over Fq, 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 {P3}. As an application, we give a refinement of the results of the finite Gan-Gross-Prasad problem in {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 {BZSV}.