Lecture: Smooth representations for p-adic groups 

Representation theory is the study of group actions on vector spaces. In the area of p-adic representation theory the groups in question are algebraic groups,

for example  GLn(F), Sp_{2n}(F), O_n(F) and SL_n(F) defined over a non-archimedean local field F. 

Because of the nature of the valuation on F these groups are totally disconnected and locally compact, and taking the algebraic group sturcture into account it is possible to approach the classification of their smooth representations.

The p-adic representation theory is part of number theory as it is connected to the smooth representations of the absolute Galois group of F via the local Langlands correspondence. 

We are going to consider the following topics during this course:

1. Classical groups and smooth representations,

2. The classification of irreducible smooth representations of GL_2(F),

3. The local Langlands correspondence for GL_2(F), 

4. The representation theory for classical groups, e.g. U(2,1)(E|F),  and the special linear group SL_2(F). 

Classical groups provide a rich class of examples to understand the obstacles of the theory. 

Literature:

1. Bushnell, Henniart: Local Langlands conjecture for GL(2) (main source)
2. Renard: Representations des groupes reductifs p-adic.
3. Serre: Linear representations of finite groups.
4. Ye Yangbo, Tian Ye p-adic Representations, Theta-Corespondence and the Langlands-Shahidi Theory (Lecture notes on a workshop in Beijing)
5. Kurinczuk: Smooth ℓ-modular representations of unramified p -adic U(2, 1)(E/F)

Exercise sheets: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5

 Sheet 6 (6.2: extra condition added) Sheet 7 Sheet 8 Sheet 9 Sheet 10 Sheet 11

Solutions (growing file  corrected and revised version)

Lecture notes are available here. Chapter III (29-86) is based on notes kindly provided by Prof. S. Stevens (UEA).