IRREDUCIBLE REPRESENTATIONS OF GL_{2}(𝔽_{q})

MICHAEL A. HILL

1. Preliminaries

The group G = GL_{2}(𝔽_{q}) is not only the slightly overweight cousin of GL_{2}(ℂ) (it is a “Flabby” S_{2}) but also a tremendously interesting finite group. It is both the automorphism group of the vector space of dimension 2 over the field 𝔽_{q} and a finite group of order q(q − 1)(q^{2} − 1), so naturally, representation theorists would like very much to understand its representations. We follow the exposition in [1].

Because the representations of GL_{2}(𝔽_{q}) 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 [ \begin{matrix}a & b \\ 0 & 1 \end{matrix} ], 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 𝔽_{q}^{+}, the group U of matrices of the form [ \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ].

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 GL_{2} 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 α_{i} are equal (and the minimal polynomial is not the characteristic polynomial), then A is α_{i} · I

If the roots α_{i} are equal and the minimal polynomial is the characteristic polynomial, then A is conjugate to [ \begin{matrix}α_{1} & 1 \\ 0 & α_{1} \end{matrix} ]

If the roots α_{i} are unequal, then A is conjugate to [ \begin{matrix}α_{1} & 0 \\ 0 & α_{2} \end{matrix} ]

If the roots are not in the field 𝔽_{q}, then they are in the unique quadratic extension 𝔽_{q^{2}}. If ¯· denotes the conjugate of an element of 𝔽_{q^{2}} over 𝔽_{q}, then N(α) = αᾱ ∈ 𝔽_{q},

and Tr(α) = α + ᾱ ∈ 𝔽_{q}, and clearly, A is conjugate to [ \begin{matrix}0 & − N(α) \\ 1 & Tr(α) \end{matrix} ], α ∈ 𝔽_{q^{2}} − 𝔽_{q}.

So, how many conjugacy classes are there? Clearly, any given matrix in G falls in one of the above conjugacy classes, so we need only find the number of conjugacy classes of each type. There are q − 1 conjugacy classes of the first two types (1 for each possible α ∈ 𝔽_{q}^{×}), \frac{1}{2}(q − 1)(q − 2) conjugacy classes of the third type, and \frac{1}{2}(q^{2} − q) of the fourth type. Thus, there are exactly q^{2} − 1 conjugacy classes.

Now that we know the number of representations, we can begin finding them. The easiest place to start is with the representations of B. Since B = Z × P, we can find the irreducible representations of B by finding those of Z and P. Z is isomorphic to 𝔽_{q}^{×}, the cyclic group of order q − 1, and so it has q − 1 irreducible representations ρ_{i}: 𝔽_{q}^{×} → ℂ^{×}. P clearly also has q − 1 1-dimensional irreducible representations ρ^{j} = ρ_{j} ∘ det. Thus, B has (q − 1)^{2} 1-dimensional characters ρ_{i}^{j} = ρ_{i} ⊗ ρ^{j}.

P has one more irreducible representation. Fix a non-trivial representation ψ: 𝔽_{q}^{+} → ℂ^{×}, and for each a ∈ 𝔽_{q}^{×}, define a representation of U by

ψ_{a}(u) = ψ(αuα^{−1}), α = [\begin{matrix}a & 0 \\ 0 & 1 \end{matrix}]

The representation ψ induces a representation on P which satisfies the interesting property that

Res_{U}^{P} Ind_{U}^{P}(ψ) = ⨁_{a∈𝔽_{q}^{×}} ψ_{a}

Frobenius reciprocity shows then that Ind_{U}^{P}(ψ) is irreducible, and the direct sum decomposition of the restriction shows that it is of dimension q − 1. Thus,

ϕ = Ind_{U}^{P}(ψ)

is the final irreducible representation of P (the sums of the squares of the dimensions of the irreducible representations is |P |).

We lift this representation to B to find the final q − 1 representations of B:

ϕ_{i} = ρ_{i} ⊗ ϕ

2. Inducing Representations of G from B

First, we observe that [G : B] = q + 1, so

dim(Ind_{B}^{G}(ρ_{i}^{j})) = q + 1

We begin by associating to each 1-dimensional representation σ of B an ordered pair of 1-dimensional representations of 𝔽_{q}^{×} (σ_{1}, σ_{2}), such that

σ ([\begin{matrix}a & ∗ \\ 0 & b \end{matrix}]) = σ_{1}(a)σ_{2}(b)

That this can be done is immediate from the fact that B/U = D and D ≅ 𝔽_{q}^{×} ×𝔽_{q}^{×}. For each σ = (σ_{1}, σ_{2}), we also define σ_{w} by

σ_{w} = (σ_{2}, σ_{1})

Thus,

σ_{w}(d) = σ(wdw^{−1}) ∀d ∈ D

Using this characterization of a 1-dimensional representation, we analyze the induced representations. Since it provides a very nice classification tool, we construct the Jacquet Module of a representation ρ of G:

J(V_{ρ}) = {v ∈ V_{ρ}|ρ(u)(v) = v, ∀u ∈ U}

Clearly,

J (⨁_{n=1}^{m} V_{ρ_{n}}) = ⨁_{n=1}^{m} J(V_{ρ_{n}})

Theorem 1. If ρ̂ is induced from a 1-dimensional representation ρ of B, then dim(J(Vρ̂)) = 2.

The proof of this is actually surprisingly simple, and since the line of reasoning is fairly indicative of the reasoning used for most theorems pertaining to representations of GL_{2}(𝔽_{q}), we present it here:

Proof. Vρ̂ is the set of all functions f: G → ℂ satisfying

f(bg) = ρ(b)f(g), ∀b ∈ B, g ∈ G.

If a vector f is in J(Vρ̂), then f also satisfies the relation

f(bu) = f(b)

Thus,

f(b) = ρ(b)f(1) and f(bwu) = ρ(b)f(w)

The Bruhat decomposition of G = B ∐(BwU) shows that any f is determined uniquely by its value on 1 and w. Thus, J(Vρ̂) = ⟨f_{1}, f_{w}⟩, where f_{1}(1) = 1 f_{1}(w) = 0, f_{w}(1) = 0 f_{w}(w) = 1. ⊠

This theorem, in conjunction with the following, shows that Ind_{B}^{G}(ρ_{i}^{j}) has at most two irreducible components.

Theorem 2. J(Vρ̂) ≠ 0 iff ρ is a direct summand of Ind_{B}^{G}(ρ_{i}^{j}) for some i and j.

A simple dimension argument shows that Ind_{B}^{G}(ρ_{i}^{j}) has at most 2 irreducible components:

Proof. Assume ρ = Ind_{B}^{G}(ρ_{i}^{j}) = ρ_{1}⊕ρ_{2}⊕⋯⊕ρ_{n}. Since each ρ_{i} is a direct summand of ρ, we know that J (V_{ρ_{i}}) ≠ 0. Thus,

J (V_{ρ}) = J (V_{ρ_{1}}) ⊕ ⋯ ⊕ J (V_{ρ_{n}})

has dimension greater than or equal to n. However, by the previous theorem, dim (J (V_{ρ})) = 2, so n ≤ 2 ⊠

The introduction of the description of a representation of B by an ordered pair of representations of 𝔽_{q}^{×} can now be used to determine exactly when a representation has 2 irreducible components:

Theorem 3. If ρ is a 1-dimensional representation of B, then the q+1 dimensional representation Ind_{B}^{G}(ρ) is reducible iff ρ = ρ_{w}. If it is reducible, then it has a 1dimensional component and an irreducible q-dimensional component.

The proof is lengthy and therefore will be omitted.

This amazing theorem not only classifies Ind_{B}^{G}(ρ) for all 1-dimensional representation of B but also tells us how many there are of each dimension:

q − 1  1-dimensional representations (ρ_{i} ∘ det)
q − 1  q-dimensional representations
\frac{1}{2}(q − 1)(q − 2) (q + 1)-dimensional representations (Ind_{B}^{G}(ρ) = Ind_{B}^{G}(ρ_{w})).

3. Cuspidal Representations

The cuspidal representations of G are those whose Jacquet module is the zeromodule. A more useful characterization is that ρ is cuspidal iff

Res_{P}^{G}(ρ) = ϕ

There are \frac{1}{2}(q^{2} − q) cuspidal representations, so simple calculation shows that if ρ is cuspidal, then

dim(ρ) = q − 1

We will use these two facts extensively in the construction of the cuspidal representations of G.

We also, in order to construct representations easily, need the “generators and relations” description of G:

Let

ω = [\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}] , z = [\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}] , s = ωz

Then

G = ⟨B, ω⟩

subject to the relations

1. ω [ \begin{matrix}a & 0 \\ 0 & b \end{matrix}] = [ \begin{matrix}b & 0 \\ 0 & a \end{matrix}] ω;

2. ω^{2} = [ \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ];

3. s^{3} = 1.

Because the cuspidal representations are difficult to describe, we must approach them via a fairly circuitous route. We describe a certain class of 1-dimensional representations of 𝔽_{q^{2}}^{×}, the non-decomposable representations, and find a way to construct a representation that corresponds to each of these.

3.1. Non-decomposable 1-dimensional Representations of 𝔽_{q^{2}}^{×}. 𝔽_{q^{2}}^{×} is a cyclic group of order q^{2} − 1, so it has q^{2} − 1 1-dimensional representations ν. We say that a representation ν of 𝔽_{q^{2}}^{×} is decomposable iff

ν(α) = ρ_{i}(N(α)).

Relatively simple computation shows that ν is decomposable if and only if ν(α) = ν(ᾱ).

In order to describe the cuspidal representations, we associate to each nondecomposable representation a cuspidal representation.

Since the dimension of the cuspidal representations is q − 1, we may pick any (q − 1)-dimensional space. It is convenient to use the space V of all functions from 𝔽_{q}^{×} into ℂ. We also want the restriction of the representation to P to be ϕ. With this in mind, we begin defining a representation ρ_{ν}.

ρ_{ν} ([\begin{matrix}a & b \\ 0 & 1 \end{matrix}]) (f)(x) = ψ(bx)f(ax)

Furthermore, it is nice if ρ_{ν} coincides with ν on Z:

ρ_{ν} ([\begin{matrix}d & 0 \\ 0 & d \end{matrix}]) (f)(x) = ν(d)f(x)

Thus,

ρ_{ν} ([\begin{matrix}a & b \\ 0 & c \end{matrix}]) (f)(x) = ν(c)ψ(bc^{−1}x)f(ac^{−1}x)

Under this definition, ρ_{ν} is a homomorphism from B into GL_{2}(V).

The difficulties arises when we try to define ρ_{ν}(ω) in such a way that ρ_{ν} preserves the relations on G. We first define a function j: 𝔽_{q}^{×} → ℂ by

j(u) = \frac{1}{q} ∑_{N(t)=u} ψ(t + t̄)ν(t)

With this defined, we can define ρ_{ν}(ω) to be

ρ_{ν}(ω)(f)(x) = ∑_{y∈𝔽_{q}^{×}} ν(y^{−1})j(xy)f(y)

That ρ_{ν} preserves the relations 1–3 is tedious, and we shall therefore only show that it satisfies relation 1.

ρ_{ν} (ω [\begin{matrix}a & 0 \\ 0 & b \end{matrix}]) f(x) = ρ_{ν}(ω)(ν(b)f(ab^{−1}x)) =

∑_{y∈𝔽_{q}} ν(by^{−1})j(xy)f(ab^{−1}y).

Making the substitution z = ab^{−1}y, we transform the sum into

∑_{z∈𝔽_{q}} ν(az^{−1})j(ba^{−1}xz)f(z) =

(ρ_{ν} ([\begin{matrix}b & 0 \\ 0 & a \end{matrix}])) ( ∑_{z∈𝔽_{q}} ν (z^{−1}) j (xz) f(z)) = ρ_{ν} ([\begin{matrix}b & 0 \\ 0 & a \end{matrix}] ω) f(x).

Since we know how ρ_{ν} acts on B and on ω, we know how it acts on G. Just for the interested reader, we present how ρ_{ν} acts on an arbitrary element in BωU:

[\begin{matrix}a & b \\ c & d \end{matrix}] = [\begin{matrix}b − ac^{−1}d & −a \\ 0 & −c \end{matrix}] [\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}] [\begin{matrix}1 & c^{−1}d \\ 0 & 1 \end{matrix}]

So,

ρ_{ν} ([\begin{matrix}a & b \\ c & d \end{matrix}]) = ∑_{y∈𝔽_{q}^{×}} (−\frac{1}{q}ψ (\frac{ax + dy}{c} ) ∑_{N(u)=xy^{−1} det(g)} ψ (−\frac{y}{c} (Tr(u))) ν(u))

By construction, Res_{P}^{G}(ρ_{ν}) = ϕ, so ρ_{ν} is cuspidal. Routine, but messy, analysis also shows that ρ_{ν} ≅ ρν̂ iff ν(α) = ν̂(ᾱ). Thus, there are exactly \frac{1}{2}(q^{2}−q) irreducible representations “induced” by non-decomposable representations.

4. Irreducible Representations of G

We present the irreducible representations in a clear, easy to read format. If we let ρ_{i}^{j} be the representation of B correlating to the ordered pair (ρ_{i}, ρ_{j}), then G has:

(q − 1)  1-dimensional representations, given by ρ_{i} ∘ det;
\frac{1}{2}(q^{2} − q)  (q − 1)-dimensional representations ρ_{ν};
(q − 1)  q-dimensional representations σ_{i}, given by σ_{i} ⊕ ρ_{i} ∘ det = Ind_{B}^{G}(ρ_{i}^{i});
\frac{1}{2}(q − 1)(q − 2) (q + 1)-dimensional representations given by Ind_{B}^{G}(ρ_{i}^{j}).

References

1. Ilya Piatetski-Shapiro, Complex representations of GL(2, K) for finite fields K, Contemporary Mathematics, vol. 16, pp. 1–43, American Mathematical Society, Providence, RI, 1980.