Chapter 3
Construction of the cuspidal representations

3.1. Projective Representations and Central Extensions

Let G be a finite group and let be a Hilbert space. Denote by U() the group of unitary automorphisms of . Let U(1) denote the group {z C : ∣z∣ = 1} under multiplication.

Definition 3.1 (Projective representation). A projective representation of G on is a function η : G → U() such that for every g,h G, there exists a constant c(g,h) U(1) such that

(3.1)

A projective representation where c(g,h) = 1 is a representation in the sense of Section 1.1 and, for emphasis, will be called an “ordinary representation”.

Exercise 3.2. Use the associative law on G to show that the function c : G × G → U(1) defined above satisfies the cocycle condition:

(3.2)

It is natural to ask whether, given a projective representation η, is it possible to find suitable scalars s(g) U(1) for each g G such that η(g)s(g) is an ordinary representation. If such a set of scalars did exist, it would mean that

Observe that

Definition 3.5 (Central Extension). A central extension of G by U(1) is a group , together with a short exact sequence

such that U(1) is contained in the centre of .

Given a central extension of G by U(1), pick any function s : G → (which may not be a homomorphism) such that the image of s(g) in G is again g. Such a function is called a section. The failure of s to be a homomorphism is measured by

(3.3)

Thus a central extension of G by U(1) determines a well-defined element of H²(G,U(1)).

Exercise 3.7. Given a cocycle c : G × G → U(1) satisfying (3.2), show that G(c) = G × U(1) with multiplication defined by

is a central extension of G by U(1). Moreover if c′ : G × G → U(1) is another cocycle, there is an isomorphism α : G(c) → G(c′) such that the diagram commutes if and only if c′c-1 is a coboundary.

In this way, H²(G,U(1)) classifies the central extensions of G by U(1). Thus, H²(G,U(1)) arises in two different contexts:

1. It measures the obstruction to modifying a projective representation to an ordinary representation.
2. It classifies the central extensions of G by U(1).

The two are related in the following way:

In other words, every projective representation can be resolved into an ordinary representation of the central extension corresponding to its cocycle.

3.2. The Heisenberg group

Assume that the finite group G is abelian. Let L²(G) denote the Hilbert space obtained when the space of complex valued functions on G is endowed with the Hermitian inner product ∑ _xf(x)g(x). On L²(G), there are two natural families of unitary operators:

Translation operators:

(Txf)(y) = f(y - x),

x G,

Modulation operators:

(Mχf)(y) = χ(y)f(y),

χ .

The translation operators give a unitary representation of G on the Hilbert space L²(G). The modulation operators give a unitary representation of on the same space. However, these operators do not commute:

Exercise 3.9. Show that

The commutator is a scalar. Thus the map η : G× → U(L²(G)) defined by

Exercise 3.10. Show that the cocycle of G× with coefficients in U(1) associated to η in (3.1) is given by

(3.4)

Explicitly, H(G) is the group whose underlying set of points is G × × U(1) with multiplication given by

(3.5)

The projective representation η of G × gives rise to an ordinary representation of H(G) on L²(G), known as the Heisenberg representation (see Exercise 3.8). Explicitly, the Heisenberg representation is realized as

(3.6)

Let θ : N → C^* be the character given by θ(0,χ,z) = z. Then the induced representation θ^H(G) is a representation of H(G) on the space

(3.7)

The action of H(G) on I is given by g′f(g) = g(gg′). For each f I, define (x) = f(-x,0,1). Since the elements (-x,0,1), with x G form a complete set of representatives of the cosets in N\H(G), f is an isomorphism of I onto L²(G). Let g′ = (x′,χ′,z′) be an element of H(G)

Let : → C^* be the character given by (x,0,z) = z. Then ^H(G) is a representation of H(G) on the space

Exercise 3.14. Show that in this realization of ^H(G) on L²(), the action of H(G) is given by

Exercise 3.15. Show that the Fourier transform FT : L²(G) → L²() defined by

is an isomorphism of H(G)-representations.

Proof. The irreducibility of follows from the following exercise:

Exercise 3.17. Use Corollary 1.18 to show that θ^H(G) is irreducible.

Suppose that ρ is an irreducible representation of H(G) on which Z acts by the identity character of U(1). By Proposition 1.19,

where (ρ) consists of a single H(G)-orbit of characters of N. By hypothesis, the restriction of all these characters to Z is the identity character of U(1).

Exercise 3.18. Show that H(G) acts transitively on the set of characters of N₁ whose restriction to Z is the identity character of U(1).

Exercise 3.19. Show that H(G)_θ = N.

Therefore, θ (ρ), and by Proposition 1.20, ρθ^H(G). □

Given an automorphism σ of H(G), let ^σ denote the representation of H(G) on the representation space V _η of η given by ^σ (g) = (^σ-1 g). If σ fixes every element of Z, then ^σ is also an irreducible representation of H(G) on which Z acts by the identity character of U(1). By Theorem 3.16, and ^σ are equivalent. Therefore, there exists ν(σ) : V _η → V _η such that

(3.8)

Moreover, by Schur’s lemma, ν(σ) is uniquely determined modulo a scalar. Let B₀(G) denote the group of all automorphisms of H(G) which fix the elements of Z.

Exercise 3.20. Show that

Conclude that ν(σ′σ) and ν(σ) ∘ ν(σ′) agree up to multiplication by a scalar.

It follows that the map σρ(σ) = ν(σ-1) is a projective representation of B₀(G) on L²(G). Projective representations of subgroups of B₀(G) constructed in this way are known as Weil representations. In order to construct ν(σ) it is helpful to think of the realization of as θ^H(G). The underlying vector space is the subspace I (see (3.7)) of C[H(G)]. Let r denote the representation of H(G) on C[H(g)], where H(G) acts by

Exercise 3.21. If f I, show that the function ν(σ)f defined by

(3.9)

is also in I. The solution will use the fact that σ fixes every element of Z. Show that ν(σ) defined above satisfies (3.8).

Exercise 3.22. Let Q : G × → U(1) denote the map

Let σ be any automorphism of G × such that Then the function : H(G) → H(G) defined by is an automorphism of H(G).

Exercise 3.23 (Symplectic form of the Heisenberg group). Assume that x2x is an automorphism of G. Consider the bijection φ : H(G) → G × × U(1) given by

The multiplication map m : H(G)² → H(G) gives rise to a new multiplication map m′ : (G××U(1))² → G××U(1) determined by the commutativity of the diagram Show that

(3.10)

3.3. A special Weil representation

In this section SL₂(Fq) will be realized as a subgroup of B₀(G) for G = Fq2. The resulting Weil representation will turn out to be an ordinary representation (Proposition 3.26). All the cuspidal representations of GL₂(Fq) and SL₂(Fq) will be found inside this representation in Sections 3.5 and 3.6 respectively. Let G be the additive group of Fq2. The map x(yψ(tr(xy))) defines an isomorphism of Fq2 onto by Proposition B.11. Using this identification, the Heisenberg group H(Fq2) can be realized as Fq2 × Fq2 × U(1), with multiplication

(3.11)

of the Heisenberg group H(G) in its usual coordinates.

Exercise 3.24. Show that in the action defined by (3.11), t(a) = , when a Fq^*, acts by

w = acts by and u(c) = , when c Fq, acts by

In the present context, (3.9) gives

(3.12)

Exercise 3.25. Show that, for any L²(G),

We have already seen that ρ : SL₂(Fq) → GL(L²(Fq2)) is a projective representation. Let be the modification of ρ by scalars given by

(3.13)

Proof. Suppose σ = , σ′ = , and σ′′ = are elements of SL₂(Fq) such that σ′′ = σσ′. Let 1₀ L²(Fq2) denote the indicator function of {0}. In the case that b, b′ and b′′ are all non-zero, we have

On the other hand, Therefore,

Now the norm map N : Fq2^*→ Fq^* is surjective, and takes each value q + 1 times (Exercise B.10). Therefore, as y ranges over Fq2^*, b′′b-1b′-1N(y) ranges over Fq^* (q + 1) times. We get

Therefore,

We already know that (σ′′) and (σ) (σ′) differ by a scalar multiple. It follows from the above calculations that this scalar multiple is 1.

If b and b′ are non-zero, but b′′ = 0, then d′b′-1 + ab-1 = 0, and the expression (3.14) equals 1, which is also the value of (σ′′)1₀(0). Again, it follows that (σ′′) = (σ) (σ′).

When exactly one of b and b′ is 0, then b′′0. In these cases, (σ) (σ′) = (σσ′) = -. □

Exercise 3.27. For a Fq^*, let t(a) = , let w = and for c Fq, let u(c) = . Use (3.13) to show that for every L²(Fq2),

(3.15) (3.16) (3.17)

Here, the Fourier transform of L²(Fq2) is once again thought of as a function of Fq2, since Fq2 is identified with its Pontryagin dual. Explicitly,

3.4. The degrees of cuspidal representations

In Chapter 2 we constructed all the representations (π,V ) of GL₂(Fq) for which

(3.18)

Representations (π,V ) satisfying (3.18) are known as the cuspidal representations of GL₂(Fq). By Frobenius reciprocity (Section 1.3), we have

Given a representation (π,V ) of any group G, let V ^* be the dual space Hom_C(V,C) of V . Let π^* be the representation of G on V ^* given by

Proposition 3.29. A representation (π,V ) of GL₂(Fq) is cuspidal if and only if there exists no non-zero vector ξ V ^* such that

(3.19)

Proof. Suppose (π,V ) is not cuspidal. Then there exists a non-zero element ξ Hom_B(V,χ) for some χ : B → C^* such that χ_∣N ≡ 1. Such a ξ can be regarded as an element of V ^*. We have, for any n N and v V ,

so that ξ satisfies (3.19).

Conversely, look at the space V ^*N of all vectors in V ^* satisfying (3.19). This space is preserved under the action of T (since tNt-1 = N for all t T). Therefore, one can write

where V _χ^*N is the space of vectors v V ^*N which transform under T by χ. If V ^*N0, then there exists χ such that V _χ^*N0. Therefore, Hom_B(V,χ)0, from which it follows that (π,V ) is not cuspidal. □

Proof. Suppose that (π,V ) is a cuspidal representation. For each a Fq, let V _a^* be the space of all ξ V ^* such that

Then the map is an isomorphism of V ^*(a) with V ^*(ta) for all t Fq^*. Hence for a0, the q - 1 spaces V ^*(ta), with t Fq^* have the same dimension. The space V ^*(0) is just V ^*(N), hence is trivial. Therefore the dimension of V ^*, hence the degree of V must be a multiple of q - 1. □

From Corollary 3.31 and the discussion at the end of Section 2.3 it follows that besides the representations constructed in that section, there are exactly (q² - q) irreducible cuspidal representations, each of degree q - 1. These representations are constructed in Section 3.5.

A cuspidal representation of SL₂(Fq) can be defined in a similar manner. A representation (π,V ) of SL₂(Fq) is said to be cuspidal if

However, Corollary 3.31 does not hold as stated

3.5. Construction of cuspidal representations of GL₂(Fq)

Let ω be a character of Fq2^* such that ωχ ∘ N for any character χ of Fq^* (here N denotes the norm map Fq2 → Fq). Such a character is called primitive.

Let

Define

Therefore, gives a representation (π_ω,W_ω) for each such ω. For any x Fq2, the set of elements x′ such that N(x′) = N(x) coincides with the set of elements of the form x′′x, where x′′ (Fq2^*)₁. Hence, if f W_ω, then the value of at x determines the value of at any element x′ with N(x′) = N(x). However, if x = 0, there is an additional constraint, namely that (0) = ω(y)-1 (0) for every y (Fq2^*)₁. By Exercise 3.35, if ω is primitive, then it is forced that (0) = 0. Since there are q - 1 non-zero values for the norm, we have

Each matrix σ in GL₂(Fq) can be written in a unique way as a product of and a matrix in SL₂(Fq). Define

(3.20)

where ã Fq2^* is chosen so that N(ã) = a.

Extend π_ω to GL₂(Fq) by = (σ). For this extended function to be a homomorphism of groups, it is necessary that, for all a,a′ Fq^* and all σ,σ′ SL₂(Fq),

(3.21)

But

′

Exercise 3.39. Using this to expand both sides of (3.21) in terms of (3.20), show that it is sufficient to check that for each a Fq^*, f L²(Fq2) and each element σ of SL₂(Fq),

(3.22)

We will denote again by (π_ω,W_ω) the restriction of to the subspace W_ω.

Proof. We will show that W_ω contains no non-zero vectors fixed by N, the subgroup consisting of matrices of the form , c Fq. This suffices, for is fixed by N if and only if π_ω(w) is fixed by N. Suppose that ₀ is a vector fixed by N. By Lemma 3.37, ₀(0) = 0. On the other hand, if x Fq2^*, then choose c Fq so that ψ(cN(x))1. Then, by (3.17)

we have ₀(x) = 0. □

Clearly, any sub-representation of a cuspidal representation is also cuspidal. Therefore, by Corollary 3.31 (π_ω,W_ω) is simple for each ω of the type considered above.

Proof. For each u Fq^*, fix an element ũ Fq2 such that N(ũ) = u. Let 1_u W_ω be the unique function such that 1_u(ũ) = 2 and 1_u(x) = 0 if N(x)u. The set {1_u ∣ u Fq^*} is a basis of W_ω. Therefore, for any σ GL₂(Fq), tr(π_ω(σ)) = ∑ _uFq^*(π_ω(σ)1_u)(ũ).

For any a Fq2^*, = . From (3.13) and (3.20), we have that

Therefore,

Exercise 3.43. Show that if ω and η are two characters of Fq2^*, then their restrictions to Fq^* are equal if and only if either ω = η or ω = η ∘ F.

If (π_ω,W_ω) and (π_η,W_η) were isomorphic, then we would have

which by Exercise 3.43 would mean that either ω = η or ω = η ∘ F. □

3.6. The cuspidal representations of SL₂(Fq)

Let ω be a non-trivial character if (Fq^*)₁, the subgroup of Fq^* consisting of elements of norm one (there are exactly q such characters). As in section 3.5 define

We shall analyze the representations π_ω through their characters. We already know that tr(π_ω()) = -1 from the proof of Lemma 3.43.

Lemma 3.45. For every character ω of (Fq2^*)₁ and d Fq such that λ² - dλ + 1 is irreducible with roots z and z-1 in Fq2,

Proof. By (3.13), we have

Using the notation of Lemma 3.43, we have Now, 1_u(y) = 0 unless y = zũ for some z (Fq2^*)₁. We have

Therefore,

If dz + z-1, then On the other hand, if d = z + z-1, then Therefore,

□

These representations must be irreducible of degree by Exercise 3.33. Using the book-keeping at the end of Section 2.5, we see that there remain irreducible representations of SL₂(Fq).

It follows that π_ω, ω non-trivial and different from the character considered in Exercise 3.46 give the remaining irreducible representations of SL₂(Fq).