Construction of the cuspidal representations

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

- The abelian group of 2-cocycles of G in U(1) consists of
functions c : G × G → U(1) which satisfy (3.2). This group is
denoted Z
^{2}(G,U(1)). - Given a function s : G → U(1), its coboundary is defined as the
cocycle c(g,h) = s(g)-1s(h)-1s(gh). The subgroup of Z
^{2}(G,U(1)) consisting of all coboundaries is denoted B^{2}(G,U(1)). - The second cohomology group of G with coefficients in U(1) is
the quotient H
^{2}(G,U(1)) = Z^{2}(G,U(1))∕B^{2}(G,U(1)).

Observe that

Proposition 3.4. For any projective representation η of G, there exists a function s : G → U(1) such that η(g)s(g) is an ordinary representation if and only if the the cocycle defined by (3.1) is a coboundary.

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

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) |

Exercise 3.6. Show that c(g,h) defined in (3.3) satisfies the cocycle condition (3.2). Moreover, if s is replaced by another section s′, and c′ is the resulting cocycle, then c′c-1 is a coboundary.

Thus a central extension of G by U(1) determines a well-defined element
of H^{2}(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

In this way, H^{2}(G,U(1)) classifies the central extensions of G by U(1).
Thus, H^{2}(G,U(1)) arises in two different contexts:

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

The two are related in the following way:

Exercise 3.8. If η is a projective representation and c is the cocycle associated to it by (3.1), then : G(c) → U() defined by (g,z) = zη(g) defines an ordinary representation of G(c).

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

Assume that the finite group G is abelian. Let L^{2}(G) denote the Hilbert
space obtained when the space of complex valued functions on G is endowed
with the Hermitian inner product ∑
_{x}f(x)g(x). On L^{2}(G), there are two
natural families of unitary operators:

Translation operators: | (T_{x}f)(y) = f(y - x), | x G, |

Modulation operators: | (M_{χ}f)(y) = χ(y)f(y), | χ . |

The translation operators give a unitary representation of G on the Hilbert
space L^{2}(G). The modulation operators give a unitary representation of
on the same space. However, these operators do not commute:

The commutator is a scalar. Thus the map η : G× → U(L^{2}(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) |

Definition 3.11 (Heisenberg group). The Heisenberg group H(G) of G is the central extension of G × by U(1) corresponding to the cocycle (3.4) (see Exercise 3.7).

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 L^{2}(G), known as the Heisenberg representation
(see Exercise 3.8). Explicitly, the Heisenberg representation is realized
as

| (3.6) |

Remark 3.12. In the construction, and in all arguments relating to the Heisenberg group H(G), where G is a finite abelian group, U(1) can be replaced by an appropriate finite subgroup. Therefore, we may pretend that H(G) is a finite group.

Exercise 3.13. Verify that N := {0}××U(1) and := G×{0}×U(1) are normal subgroups of H(G). Z := {0}×{0}×U(1) is the centre of H(G). Here 0 denotes the identity element of either G or .

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 L^{2}(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 3.15. Show that the Fourier transform FT : L^{2}(G) → L^{2}()
defined by

Theorem 3.16. The representation is irreducible. Every irreducible representation of H(G) on which Z acts by the identity character of U(1) is isomorphic to .

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,

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

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 B_{0}(G) denote the group of all automorphisms of H(G) which fix the
elements of Z.

It follows that the map σρ(σ) = ν(σ-1) is a projective representation of
B_{0}(G) on L^{2}(G). Projective representations of subgroups of B_{0}(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

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

| (3.10) |

In this section SL_{2}(Fq) will be realized as a subgroup of B_{0}(G) for G = Fq2.
The resulting Weil representation will turn out to be an ordinary
representation (Proposition 3.26). All the cuspidal representations
of GL_{2}(Fq) and SL_{2}(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

In the present context, (3.9) gives

| (3.12) |

We have already seen that ρ : SL_{2}(Fq) → GL(L^{2}(Fq2)) is a projective
representation. Let be the modification of ρ by scalars given by

| (3.13) |

Proposition 3.26. The function : SL_{2}(Fq) → GL(L^{2}(Fq)) defined
by (3.13) is an ordinary representation.

Proof. Suppose σ = , σ′ = , and σ′′ = are elements
of SL_{2}(Fq) such that σ′′ = σσ′. Let 1_{0} L^{2}(Fq2) denote the indicator
function of {0}. In the case that b, b′ and b′′ are all non-zero, we
have

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′′)1_{0}(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 L^{2}(Fq2),

(3.15) (3.16) (3.17) |

Exercise 3.28. Any element of SL_{2}(Fq) can be written as a
product of elements of the above types. Consider the matrix
SL_{2}(Fq). If b = 0, then d = a-1 and = t(a)u(ac). On the other hand,
if b0, then = u(d∕b)wu(ab)t(b-1).

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

| (3.18) |

Representations (π,V ) satisfying (3.18) are known as the cuspidal
representations of GL_{2}(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 GL_{2}(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 ,

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

Corollary 3.31. The degree of every cuspidal representation of
GL_{2}(Fq) is always a multiple of (q - 1).

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

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 (q^{2} - q) irreducible cuspidal representations, each of degree
q - 1. These representations are constructed in Section 3.5.

A cuspidal representation of SL_{2}(Fq) can be defined in a similar
manner. A representation (π,V ) of SL_{2}(Fq) is said to be cuspidal
if

Exercise 3.32. Verify that Proposition 3.29 continues to hold
when GL_{2}(Fq) is replaced by SL_{2}(Fq).

However, Corollary 3.31 does not hold as stated

Exercise 3.33. Show that the degree of a cuspidal representation
of SL_{2}(Fq) is always a multiple of .

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

Exercise 3.35. Show that a character ω : Fq2^{*} → C^{*} is primitive
if and only if its restriction to (Fq2^{*})_{1} is non-trivial.

Define

Exercise 3.36. Show that W_{ω} is preserved by the action of (σ)
for every σ SL_{2}(Fq). [Hint: note that if N(x) = 1, then x = x-1.]

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 GL_{2}(Fq) can be written in a unique way as a product of
and a matrix in SL_{2}(Fq). Define

| (3.20) |

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

Exercise 3.38. Check that the right hand side of (3.20) does not
depend on the choice of ã such that N(ã) = a, and that it preserves W_{ω}
for each primitive ω.

Extend π_{ω} to GL_{2}(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′ SL_{2}(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 L^{2}(Fq2)
and each element σ of SL_{2}(Fq),

| (3.22) |

Exercise 3.40. Verify (3.22) for σ of the form t(a), w and u(c) (see
Exercise 3.28). Conclude that it holds for all σ SL_{2}(Fq).

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)

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.

Lemma 3.42. Let ω and η be two characters of Fq2^{*} as above. If the
representations (π_{ω},W_{ω}) and (π_{η},W_{η}) are isomorphic, then either ω = η
or ω = η ∘ F, where F is the Frobenius automorphism Fq2^{*} → Fq2^{*} (see
Section B.3).

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 GL_{2}(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

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

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
λ^{2} - dλ + 1 is irreducible with roots z and z-1 in Fq2,

Proof. By (3.13), we have

Exercise 3.46. Suppose that ω is the unique non-trivial character
of (Fq2^{*})_{1} taking only the values ±1. Show that ∑
_{σSL2(Fq)}tr(π_{ω}(σ)) =
2(q^{3}-q). Conclude that π_{ω}(σ) is a sum of two non-isomorphic irreducible
representations of SL_{2}(Fq).

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 SL_{2}(Fq).

Exercise 3.47. Define an equivalence relation on the set of
non-trivial characters of (Fq2^{*})_{2} by ω ~ ω′, where ω′ = ω∘F. Here F is the
Frobenius automorphism (Section B.3). Observe that tr(π_{ω}) = tr(π_{ω′}).
Show that the characters of the representations π_{ω}, where ω runs over
the equivalence classes of non-trivial characters of (Fq2^{*})_{1} are pairwise
orthogonal.

It follows that π_{ω}, ω non-trivial and different from the character considered
in Exercise 3.46 give the remaining irreducible representations of
SL_{2}(Fq).