IRREDUCIBLE REPRESENTATIONS OF GL2(Fq)

The group G = GL2(Fq) is not only the slightly overweight cousin of GL2(C) (it is a “Flabby” S2) but also a tremendously interesting finite group. It is both the automorphism group of the vector space of dimension 2 over the field Fq and a finite group of order q(q − 1)(q − 1), so naturally, representation theorists would like very much to understand its representations. We follow the exposition in [1]. Because the representations of GL2(Fq) of dimension higher than 1 are difficult to describe, we need to better describe the structure of G. Probably the most important subgroup of G is B, the Borel subgroup. B, together with the non-trivial permutation matrix w allow G to be decomposed via the Bruhat decomposition. In order to find the representations of G, we will induce representations from B. B contains the group P of “shears,” matrices of the form [ a b 0 1 ], in addition to the group D of diagonal matrices and Z = Z(G), the scalar multiples of the identity. Finally, B contains a copy of Fq , the group U of matrices of the form [ 1 a 0 1 ]. We are now in a position to begin describing the representation of G, but how many are there? This question is equivalent to the question of the number of conjugacy classes of G, so we begin by computing this. Linear algebra tells us that two matrices in GL2 are conjugate iff they satisfy the same minimal polynomial. Thus, we can determine the conjugacy classes by looking at the minimal polynomial of an arbitrary matrix. Since G is the group of 2× 2 matrices, the characteristic polynomial is the minimal polynomial whenever the matrix is not a scalar multiple of the identity, our job is actually quite easy. We classify the characteristic polynomial of a matrix A based on its roots: If the roots αi are equal (and the minimal polynomial is not the characteristic polynomial), then A is αi · I If the roots αi are equal and the minimal polynomial is the characteristic polynomial, then A is conjugate to [ α1 1 0 α1 ]