Lusztig correspondence and the Gan-Gross-Prasad problem

TL;DR

This paper completely solves the Gan-Gross-Prasad restriction problem for finite classical groups by leveraging Lusztig correspondence and Reeder's formula, extending previous work on unipotent representations to all representations.

Contribution

It introduces a reduction method using Lusztig correspondence and Reeder's formula to solve the Gan-Gross-Prasad problem for all representations of finite classical groups.

Findings

Complete solution to the Gan-Gross-Prasad problem over finite fields.

Reduction decomposition of Reeder's formula.

Extension from unipotent to all representations via Lusztig correspondence.

Abstract

The Gan-Gross-Prasad problem is to describe the restriction of representations of a classical group $G$ to smaller groups $H$ of the same kind. In this paper, we solved the Gan-Gross-Prasad problem over finite fields completely. In previous work \cite{LW1,LW2,LW3,Wang}, we study the Gan-Gross-Prasad problem for unipotent representations of finite classical groups. The main tools used are the Lusztig correspondence as well as a formula of Reeder \cite{R} for the pairings of Deligne-Lusztig characters. We give a reduction decomposition of Reeder's formula, and deduce the Gan-Gross-Prasad problem for arbitrary representations from the unipotent representations by Lusztig correspondence.