Representations of GL2(Fq) and SL2(Fq)

Representations of
GL₂(Fq) and SL₂(Fq),
and some remarks about GL_n(Fq)

Amritanshu Prasad

The Institute of Mathematical Sciences, Chennai.

E-mail address: amri@imsc.res.in

URL: http://www.imsc.res.in/~amri

Notes from a course taught at
the Advanced Instructional School on Representation Theory and Related Topics
held at the Bhaskaracharya Pratishthana and the University of Pune in July 2007

2000 Mathematics Subject Classification. 20C33

Key words and phrases. Representation theory, algebraic groups, finite fields, Weil representation

  Contents
  Introduction
Chapter 1.  General results from representation theory
1.1.  Basic definitions
1.2.  The Pontryagin dual of a finite abelian group
1.3.  Induced Representations
1.4.  Description of intertwiners
1.5.  A criterion for irreducibility
1.6.  The little groups method
Chapter 2.  Representations constructed by
parabolic induction
2.1.  Conjugacy classes in GL₂(F_q)
2.2.  Subgroup of upper-triangular matrices
2.3.  Parabolically induced representations for GL₂(F_q)
2.4.  Conjugacy classes in SL₂(F_q)
2.5.  Parabolically induced representations for SL₂(F_q)
Chapter 3.  Construction of the cuspidal representations
3.1.  Projective Representations and Central Extensions
3.2.  The Heisenberg group
3.3.  A special Weil representation
3.4.  The degrees of cuspidal representations
3.5.  Construction of cuspidal representations of GL₂(F_q)
3.6.  The cuspidal representations of SL₂(F_q)
Chapter 4.  Some remarks on GL_n(F_q)
4.1.  Parabolic Induction
4.2.  Cuspidal representations
Appendix A.  Similarity Classes of Matrices
A.1.  Basic properties of matrices
A.2.  Primary decomposition
A.3.  Structure of a primary matrix
A.4.  Block Jordan canonical form
A.5.  Centralisers
A.6.  Perfect fields
Appendix B.  Finite Fields
B.1.  Existence and uniqueness
B.2.  The multiplicative group of F_q
B.3.  Galois theoretic properties
B.4.  Identification with Pontryagin dual
  Bibliography