Classification of representations of finite groups of Lie type and the Langlands framework

This thesis investigates the representation theory of finite groups of Lie type and p-adic groups, with a focus on the classification of irreducible representations and the interplay between finite and local perspectives. The first part addresses finite groups of Lie type, focusing on the case in which the enveloping algebraic group is disconnected. Chapter 1 extends the theory of parabolically induced cuspidal representations to this setting, showing that the associated endomorphism algebras remain finite extended Hecke algebras and analyzing the restriction of characters. Chapter 2 builds on this by exhibiting a combinatorial proof of the compatibility of Lusztig’s non-abelian Fourier transform with parabolic induction for classical groups of types B, C, and D, with particular attention to disconnected groups of type D. The second part explores links with the local Langlands program. Chapter 3 establishes the compatibility between the Macdonald correspondence for the general linear group and the tame local Langlands correspondence via parahoric restriction. Chapter 4, based on a conjecture by Imai and Vogan, extends this framework to the special linear group, introducing a parameterization for the irreducible representations of the finite special linear group that can be regarded as a finite Langlands correspondence, and proving its compatibility with the tame local Langlands correspondence, thereby connecting the finite and local theories.