Introduction 1 1. The unramified global Langlands correspondence 5 2. Classical local Langlands correspondence 9 3. Geometric local Langlands correspondence over C 12 4. Center and opers 18 5. Opers vs. local systems 23 6. Harish–Chandra categories 26 7. Local Langlands correspondence: unramified case 30 8. Local Langlands correspondence: tamely ramified case 41 9. Ramified global Langlands cor...

This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [Lan70] is a conjecture of utmost importance, concerning global fields, i.e. number fields and function fields. Many excellent surveys a...

The local Langlands correspondence is a conjectural connection between representations of groups G(k) for connected reductive groups G over a p-adic field k and certain homomorphisms (Langlands parameters) from the Galois (or WeilDeligne group) of k into a complex Lie group Gwhich is dual, in a certain sense, to G and which encodes the splitting structure of G over k. More introductory remarks ...

In this paper, we survey some mathematical developments that followed from the discovery of simple supercuspidal representations p-adic groups.