The goal of these notes is to give a self-contained account of the representation theory of GL2 and SL2 over a finite field, and to give some indication of how the theory works for GLn over a finite field.
Let Fq denote a finite field with q elements, where q is a prime power. The irreducible characters of GL2(Fq) and SL2(Fq) were classified by Herbert E. Jordan [Jor07] and Issai Schur [Sch07] in 1907. The method used here is not that of Jordan or Schur, but depends on a construction known as the Weil representation introduced by André Weil in his famous article [Wei64]. Weil’s method was used to obtain all the irreducible representations of SL2(Fq) in [Tan67] by Shun’ichi Tanaka. A very readable exposition is also found in Daniel Bump’s book [Bum97, Section 4.1] in the case of GL2(Fq). These two works have been my main sources. The use of the Weil representation has the disadvantage that it does not generalise to other groups (such as GLn(Fq), SLn(Fq), or other finite groups of Lie type, with the exception of Sp4(Fq)). On the other hand, the Weil representation is important in number theory as well as representation theory. For example, a version of the Weil representation plays an important role in the construction of supercuspidal representations of reductive groups over non-Archimedean local fields, as was first demonstrated by Takuro Shintani in [Shi68]. A systematic use of the Weil representation in this context is made by Paul Gérardin in [Gér75]. These techniques have been used with considerable success to prove the local Langlands conjectures for non-Archimedean local fields, but this is a matter that will not be discussed here.
For n × n matrices, the representations were classified by James A. Green in 1955 [Gre55]. The general linear groups are special cases of a class of groups known as reductive groups, which occur as closed subgroups of general linear groups (in the sense of algebraic geometry). In 1970, T. A. Springer presented a set of conjectures describing the characters of irreducible representations of all reductive groups over finite fields, some of which he attributed to Ian G. MacDonald [Spr70]. The essence of these conjectures is that the irreducible representations of reductive groups over finite fields occur in families associated to maximal tori in these groups (in this context, a torus is a subgroup that is isomorphic to a product of multiplicative groups of finite extensions of Fq). A big breakthrough in this subject came in 1976, when Pierre Deligne and George Lusztig [DL76], were able to construct the characters of almost all the irreducible representations (in an asymptotic sense) of all reductive groups over finite fields, in particular, proving the conjectures of MacDonald. Much more information about the irreducible representations of reductive groups over finite fields has been obtained in later work, particularly by Lusztig (see e.g., [Lus84]). The above survey is far from complete and fails to mention many important developments in the subject. It is intended only to give the reader a rough sense of where the material to be presented in these lectures lies in the larger context of 20th century mathematics.
I am grateful to Pooja Singla, who carefully read an earlier version of these notes and pointed out several errors. I have had many interesting discussions with her on the representation theory of GL2(Fq), which have helped me when I wrote these notes. I am grateful to M. K. Vemuri, from whom I have learned a large part of what I know about Heisenberg groups and Weil representations.