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

Amritanshu Prasad

The Institute of Mathematical Sciences, Chennai.


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

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(Fq)
 2.2.  Subgroup of upper-triangular matrices
 2.3.  Parabolically induced representations for GL2(Fq)
 2.4.  Conjugacy classes in SL2(Fq)
 2.5.  Parabolically induced representations for SL2(Fq)
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(Fq)
 3.6.  The cuspidal representations of SL2(Fq)
Chapter 4.  Some remarks on GLn(Fq)
 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 Fq
 B.3.  Galois theoretic properties
 B.4.  Identification with Pontryagin dual