Chapter 2
Representations constructed by
parabolic induction

2.1. Conjugacy classes in GL₂(Fq)

Given a matrix in GL₂(Fq) consider its characteristic polynomial

To summarise, the conjugacy classes in GL₂(Fq) are as follows:

1. (q - 1) classes represented by , with λ Fq^*.
2. (q - 1) classes represented by , with λ Fq^*.
3. (q - 1)(q - 2) classes represented by with λ₁λ₂.
4. (q² - q) classes represented by , with λ² - a₁λ + a₀ an irreducible polynomial in Fq[t].

In all, there are

(2.1)

conjugacy classes. Detailed information about the conjugacy classes is collected in Table 1.

Table 1: Conjugacy classes of GL2(Fq) Name element centraliser no. of classes size of class Central , a Fq* q(q - 1)2(q + 1) q - 1 1 Non-semisimple , a Fq* q(q - 1) q - 1 (q - 1)(q + 1) Split regular semisimple ab Fq* (q - 1)2 q(q + 1) Anisotropic regular semisimple Cp, p(t) Fq[t] quadratic, irreducible (q - 1)(q + 1) q(q - 1)

2.2. Subgroup of upper-triangular matrices

Let B be the subgroup of GL₂(Fq) consisting of invertible upper triangular matrices. Let N be the subgroup of upper triangular matrices with 1’s along the diagonal. Let T be the subgroup of invertible diagonal matrices.

Let w = .

Proposition 2.5 (Bruhat decomposition).

Note that B is really a double coset B1B. So Proposition 2.5 really tells us that the double coset space B\GL₂(Fq)∕B has two elements and that {1,w} is a complete set of representatives for these double cosets.

Proof. A matrix lies in B if and only if c = 0. If c0, then

□

2.3. Parabolically induced representations for GL₂(Fq)

Given characters χ₁ and χ₂ of Fq^*, we get a character χ of T by

(2.2)

Let I(χ₁,χ₂) be the representation of GL₂(Fq) induced from this character of B.

Proposition 2.6. Let χ₁, χ₂, μ₁ and μ₂ be characters of Fq^*. Then

where, and

Proof. Let χ and μ be the characters of B obtained from the pairs χ₁, χ₂ and μ₁, μ₂ respectively as in (2.2). We regard χ and μ as one-dimensional representations of B acting on the space C. We may identify Hom_C(C,C) with C as well. Then, using Mackey’s description of intertwiners (Theorem 1.12), we see that we must compute the dimension of the space of functions Δ : GL₂(Fq) → C such that

(2.3)

It follows from the Bruhat decomposition that Δ is completely determined by its values at 1 and w.

Taking g = 1 in (2.3), we see that for any t T,

Therefore, if μχ then Δ(1) = 0. On the other hand, if μ = χ, let Δ₁ be the function such that and whose restriction to BsB is zero. If e₁ = 0, we take Δ₁ ≡ 0.

Taking g = w in (2.3), we see that for any t T,

Exercise 2.7. w-1w = .

Therefore, if μ₁χ₂ or μ₂χ₁ then Δ(w) = 0. On the other hand, if μ₁ = χ₂ and μ₂ = χ₁, let Δ_w be the function such that

and whose restriction to B is 0. If e_w = 0, we take Δ_w ≡ 0.

An arbitrary functions satisfying (2.3) can be expressed as a linear combination of Δ₁ and Δ_w, so we see that the dimension of the space of such functions must be e₁ + e_w. □

Theorem 2.8. Let χ₁, χ₂, μ₁ and μ₂ be characters of Fq^*. Then I(χ₁,χ₂) is an irreducible representation of degree q + 1 of GL₂(Fq) unless χ₁ = χ₂, in which case it is a direct sum of two irreducible representations having degrees 1 and q. We have

if and only if either

(2.4)

or else

(2.5)

Proof. Apply Proposition 2.6 with χ₁ = μ₁ and χ₂ = μ₂. We see that

Recall that if (π,V ) is a representation of a finite group G and V is a direct sum of distinct irreducible representations π₁,,π_h with multiplicities m₁,,m_h and with degrees d₁,,d_h respectively, then the dimension of End_G(V ) is ∑ m_id_i². Hence I(χ₁,χ₂) is irreducible if χ₁χ₂, otherwise it is a direct sum of two irreducible representations because 2 = 1² + 1² is the only way of writing 2 as a sum of non-zero multiples of more than one non-zero squares.

Because the index of B in GL₂(Fq) is q + 1, the dimension of I(χ₁,χ₂) is always q + 1. If χ₁ = χ₂, the representation of GL₂(Fq) generated by the function f(g) = χ₁(det(g)) clearly satisfies f(bg) = χ(b)f(g) for all b B and g G. Therefore f I(χ₁,χ₂). Moreover, (g ⋅f)(x) = χ₁(det(g))f. Therefore the one-dimensional subspace spanned by f is invariant under the action of G, hence forms a one-dimensional representation of G. The other component is therefore q-dimensional.

If χ₁χ₂ then, I(χ₁,χ₂) is irreducible. By Proposition 2.6 there exists a non-zero element in Hom(I(χ₁,χ₂),I(μ₁,μ₂)) if and only if χ₁ = μ₁ and χ₂ = μ₂ or χ₁ = μ₂ and χ₂ = μ₁. By irreducibility, these homomorphisms must be isomorphisms. This proves the second part of the theorem. □

To summarise, in this section, we have constructed irreducible representations of GL₂(Fq) corresponding to characters χ = (χ₁,χ₂) of T:

1. When χ₁χ₂, there is a unique irreducible representation of GL₂(Fq) of degree q + 1 corresponding to χ; the irreducible representation corresponding to (χ₁,χ₂) is isomorphic to the one corresponding to (χ₂,χ₁). We have (q-1)(q-2) irreducible representations of degree q + 1.
2. When χ₁ = χ₂, there are two irreducible representations of GL₂(Fq) corresponding to χ, one of degree 1 and the other of degree q. All these representations are pairwise non-isomorphic. Therefore we have q - 1 representations of degree 1 and q - 1 representations of degree q.

Recall from Schur theory, that the number of irreducible representations is the same as the number of conjugacy classes in a group. We have constructed

Recall that for a group of order n whose irreducible representations are π₁,…,π_r of degrees d₁,…,d_r respectively,

The sum of squares of degrees of the representations that we have constructed so far is

We will see in Section 3.4 that there are (q² - q) irreducible representations of degree q - 1 remaining. These will be constructed in Section 3.5.

2.4. Conjugacy classes in SL₂(Fq)

Let Aut(SL₂(Fq)) denote the group of all automorphisms of SL₂(Fq). GL₂(Fq) acts on SL₂(Fq) by conjugation. This gives rise to a homomorphism GL₂(Fq) → Aut(SL₂(Fq)). The kernel of this automorphism consists of scalar matrices in GL₂(Fq), and is therefore isomorphic to Fq^*. The image is therefore isomorphic to the group PGL₂(Fq), which is the quotient of GL₂(Fq) by the subgroup of invertible scalar matrices. The orbits of PGL₂(Fq) on SL₂(Fq) are precisely the conjugacy classes of GL₂(Fq) which are contained in SL₂(Fq) (note that SL₂(Fq) is a union of conjugacy classes of GL₂(Fq)).

On the other hand, the image of SL₂(Fq) in Aut(SL₂(Fq)) is PSL₂(Fq), the quotient of SL₂(Fq) by the subgroup {±1}. The conjugacy classes of SL₂(Fq) are precisely the PSL₂(Fq) orbits.

Now, PSL₂(Fq) is a subgroup of PGL₂(Fq) (when both groups are viewed as subgroups of Aut(SL₂(Fq))) of index two. Therefore, each conjugacy class of GL₂(Fq) whose elements lie in SL₂(Fq) is either a single conjugacy class in SL₂(Fq) or a union of two conjugacy classes in SL₂(Fq). If ε is an element of Fq^* which is not a square (since q is assumed to be odd, there are such elements), then the image of in Aut(SL₂(Fq)) does not lie in the image of SL₂(Fq).

Let σ SL₂(Fq). Whether or not the conjugacy class of σ in GL₂(Fq) splits or not can be determined by counting. The basic principle here is that the number of elements in an orbit for a group action is the index of the stabiliser of a point in the orbit.

With the above observations in mind, it is not difficult to prove that

In all, there are

(2.6)

conjugacy classes.

Table 2: Conjugacy classes of SL2(Fq) Name representative centraliser no. of classes size of class Central q(q - 1)(q + 1) 2 1 Non- 2q 2 semisimple 2q 2 Split regular semisimple a Fq*\{±1} q - 1 q(q + 1) Anisotropic regular semisimple Cp, p[t] Fq[t] irreducible, p(0) = 1 q + 1 q(q - 1)

2.5. Parabolically induced representations for SL₂(Fq)

Let B now consist of the upper triangular matrices in SL₂(Fq), N the upper triangular matrices with 1’s along the diagonal, and T the matrices in SL₂(Fq) which are diagonal. Note that the results of Exercise 2.4 are still valid, as is the Bruhat decomposition:

Proposition 2.14. Let χ and μ be characters of Fq^*. Then,

where,

Taking μ = χ in Proposition 2.14 gives that

Case χ(ε) = 1

In this case, I(χ) contains the invariant one dimensional subspace of constant functions on G. Therefore I(χ) splits into a direct sum of two irreducible representations, the trivial representation and a representation of dimension q, which is called the Steinberg representation.

Case χ(ε) = -1

In this case it is necessary to make a closer analysis of End_SL2(Fq)(I(χ)). Let Δ₁ denote the unique function SL₂(Fq) → C for which Δ₁(1) = 1, Δ₁(w) = 0, and Δ(b₁gb₂) = χ(b₁)Δ(g)χ(b₂). Also let Δ_w denote the unique function SL₂(Fq) → C for which Δ_w(1) = 0, Δ_w(w) = 1, and Δ(b₁gb₂) = χ(b₁)Δ(g)χ(b₂). These two functions form a basis of End_SL2(Fq)(I(χ)). Write I(χ) = ρ⁺ ⊕ ρ^-, where ρ⁺ and ρ^- are the two irreducible summands of of I(χ). The identity endomorphism in I(χ) can be written as a sum of two idempotents, coming from the identity endomorphisms of ρ⁺ and ρ^-.

Exercise 2.17. Besides q-1(q - 1)-1Δ₁ and 0, show that the only idempotents in End_SL2(Fq)(I(χ)) are

Here κ = 0 if χ(-1) = 1 and κ = 1 if χ(-1) = -1.

Let ^εI(χ) be the representation of SL₂(Fq) on the representation space of I(χ), but where the action of SL₂(Fq) is given by

Therefore, the two representations ρ⁺ and ρ^- must have equal degrees. It follows that I(χ) is a sum of two irreducible representations, each of degree .

In this section, we have constructed