representations of gl(2;f for nite and local elds f...

Representations of GL(2, F )

for finite and local fields F

Laurence Samuel Field

An essay submitted in partial fulfillment of

the requirements for the degree of

B.Sc. (Advanced Mathematics) (Honours)

Pure Mathematics

University of Sydney

SID

EREMENS EAD

EMMUTAT

O

October 2008

Acknowledgements

I would first like to acknowledge my supervisor, Donald Cartwright, for his help over

the course of this year. Donald has opened my eyes to a wide range of interesting

mathematical ideas, and has helped me to appreciate some of the interconnectedness

of mathematics. His ability to come up with ingenious arguments at the drop of a hat

is impressive, and his attention to the progress of my work has made my Honours year

much more enjoyable.

I would especially like to thank Kate Turner for her constant loving support, her

proofreading and her many good suggestions. Her experience, having written an Hon-

ours essay last year, has been invaluable.

I would also like to thank my parents Michael and Gabrielle for their support, and

for proofreading.

Finally, I would like to thank Andrew Crisp for his friendship this year and for his

useful suggestions.

iii

Contents

1 Introduction 1

1.1 Index of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background theory 3

2.1 Representations and intertwiners . . . . . . . . . . . . . . . . . . . . . 3

2.2 Induced representations and Mackey theory . . . . . . . . . . . . . . . 5

2.3 The Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 The structure of SL(2, F ) . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Representations over finite fields 13

3.1 The Finite Stone–von Neumann Theorem . . . . . . . . . . . . . . . . . 13

3.2 A projective representation of SL(2, F ) . . . . . . . . . . . . . . . . . . 19

3.3 A true representation of SL(2, F ) . . . . . . . . . . . . . . . . . . . . . 22

3.4 A motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Extending the representation to GL(2, F ) . . . . . . . . . . . . . . . . 28

3.6 Cuspidality and irreducibility . . . . . . . . . . . . . . . . . . . . . . . 31

3.7 Inequivalence of cuspidal representations . . . . . . . . . . . . . . . . . 34

4 Representations over local fields 39

4.1 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Smooth and admissible representations . . . . . . . . . . . . . . . . . . 44

4.3 Fourier analysis on local fields . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 A projective representation of SL(2, F ) . . . . . . . . . . . . . . . . . . 47

4.5 A true representation of SL(2, F ) . . . . . . . . . . . . . . . . . . . . . 51

4.6 Extending the representation to GL(2, F ) . . . . . . . . . . . . . . . . 58

v

vi Contents

Chapter 1

Introduction

The representations of the general linear group are intrinsically interesting, and they are

also relevant to many other areas of mathematics. In particular, they are connected

to the theory of automorphic forms via an object known as the adele ring (see [1],

Chapter 3). We will not explore these connections, but will instead strive to understand

the representations of GL(2, F ) for certain fields F .

The definition and basic properties of local fields are stated in Section 4.1. It turns

out that local fields come in two classes, Archimedean and non-Archimedean. Of these

two classes, the non-Archimedean local fields are in some senses simpler, and they have

a surprising amount in common with finite fields; indeed, for each non-Archimedean

local field there exists a naturally defined residue field, which is finite. In this essay

we construct representations of GL(2, F ) where F is a finite field, and where F is a

non-Archimedean local field. It turns out that these two cases may be discussed in

a way that is similar in broad outline. Indeed, in this essay we reuse many of the

propositions from the finite field case in the non-Archimedean local field case.

For finite fields F , we construct all the irreducible representations of GL(2, F ).

They fall into four categories: characters, special representations, principal series rep-

resentations and cuspidal representations. All of these representations are given in more

or less explicit form, except for the special representations (for the sake of brevity). By

counting arguments we verify that all the irreducibles have been found.

For non-Archimedean local fields F , we construct a large family of representations

of GL(2, F ), known as dihedral supercuspidal representations, which correspond to the

cuspidal representations of the finite field case. We do not construct any other families

of representations, although a slight variant of our construction yields the principal

series representations in a way we shall not discuss (see [1], p. 543). The representations

we construct are irreducible, but the verification of this fact requires theory that is

beyond the scope of this essay (see [1], Sections 4.7 and 4.8). Because our constructions

depend heavily on the diagonalisation of symmetric bilinear forms, we shall not consider

those non-Archimedean local fields whose residue fields have characteristic 2. Indeed,

in such cases, the standard categorisation of irreducible representations breaks down

in a way described by the local Langlands conjecture (see [1], Section 4.9).

1

2 Chapter 1. Introduction

There are certainly many ways to construct representations of GL(2, F ), and the

method adopted in this essay is not the most efficient method possible. An alterna-

tive, perhaps more elementary, method to construct the irreducible representations of

GL(2, F ) for finite fields F is described by Piatetski-Shapiro [7]. The irreducible rep-

resentations of GL(2, F ) are in some sense parametrised by the characters of its tori,

a viewpoint expounded (in greater generality) by Deligne and Lusztig [3].

The route we take to construct representations of GL(2, F ) is the one outlined by

Bump [1]. Most of this essay is concerned with constructing representations of GL(2, F )

by means of the Weil representation. This is a construction introduced in 1964 in

a famous paper of Weil [10], which is applicable in considerably greater generality

than we shall consider in this essay. We give a brief treatment of the principal series

representations as induced representations in Section 3.4, but these representations are

subsequently rediscovered by means of the Weil representation in Section 3.6.

In Chapter 2, we give some basic definitions, introduce the induced representation,

develop Mackey’s theory of intertwiners, introduce the Heisenberg group, and provide

a presentation for SL(2, F ). In Chapter 3, we prove the Finite Stone–von Neumann

Theorem, and we use that as a starting point to construct representations of SL(2, F )

for finite fields F by means of the Weil representation. We arrive at a complete de-

scription of the irreducible representations of GL(2, F ). In Chapter 4, we introduce

local fields, introduce an appropriate notion of admissible representations, and proceed

to construct a large family of representations of GL(2, F ) for non-Archimedean local

fields F by means of the Weil representation.

1.1 Index of notation

The table below provides a guide to the notation and nomenclature that will be used.

Terms introduced in the text appear in small caps when they are first defined. Blank

entries in matrices always denote zero. We frequently use 0 and 1 to represent the trivial

subspace {0} and trivial subgroup {1} respectively.

Notation Meaning

N The natural numbers {1, 2, 3, . . .}I The identity matrix

A tB Disjoint union of A and B

T The group{t ∈ C

∣∣ |t| = 1}

R× The group of invertible elements of the ring R

Z(G) The centre of the group G

H1\G/H2 The set of double cosets {H1gH2 | g ∈ G}1S The characteristic function of the set S

charF The characteristic of the field F

|a|p The p-adic absolute value of a

supp Φ The support {v ∈ V | Φ(v) 6= 0} of the function Φ : V → C

Chapter 2

Background theory

In this chapter we establish the basic theoretical tools we will need to generate repre-

sentations of GL(2, F ) for finite and local fields F . We will often construct represen-

tations by the construction known as the induced representation, and Mackey theory

provides the tools we need to analyse intertwiners between induced representations. We

introduce the Heisenberg group, which we will use to construct certain projective rep-

resentations of SL(2, F ). Finally, we give a presentation of SL(2, F ), which is needed

to check that the projective representations of SL(2, F ) we construct can be lifted to

true representations.

2.1 Representations and intertwiners

Definition 2.1. Let G be a group. A representation of G is a group homomorphism

π : G→ GL(V ), for some complex vector space V . We often denote the representation

by (π, V ), to emphasise the space on which G acts. A projective representation

of G is a homomorphism ρ : G→ PGL(V ), for some complex vector space V .

Definition 2.2. A character of a topological group G is a continuous group homo-

morphism χ : G→ T. If G is finite, then we assume no topology on G, so a character

of G is in that case any homomorphism χ : G→ T.

An additive character of a ring R is a character of R considered as a group

under addition; a multiplicative character of R is a character of the group of

units R×. Any topology on R is taken to apply also to these groups.

A quasicharacter of a topological group G is a continuous group homomorphism

χ : G→ C×.

Remarks 2.3. If χ is a quasicharacter of a finite group G, then χ(g) has finite order for

all g ∈ G. Thus imχ ⊆ T, so χ is in fact a character.

Often we will regard a character of a group as a one-dimensional representation of

that group on the space C (by multiplication).

We will not use the word character to mean a map giving the traces of the repre-

senting matrices of a representation.

3

4 Chapter 2. Background theory

Definition 2.4. If (π1, V1) and (π2, V2) are representations of a group G, we say a linear

map T : V1 → V2 is an intertwiner from π1 to π2 if T ◦π1(g) = π2(g)◦T for all g ∈ G.

The space of intertwiners from π1 to π2 will be denoted HomG(π1, π2). This is not to be

confused with the (larger) space HomC(V1, V2) of vector space homomorphisms from V1

to V2. The representations π1 and π2 are equivalent if there exists an intertwiner

T : V1 → V2 from π1 to π2 which is also a vector space isomorphism.

We will often want to understand the space of all intertwiners between two given

representations, because this space gives us useful information about the structure of

those representations.

The complete reducibility of each representation of a finite group G on a finite-

dimensional complex vector space, known as Maschke’s theorem, is the key ingredient

in the following results. For the remainder of this section, we assume all representation

spaces are finite-dimensional.

Proposition 2.5. Let G be a finite group, and let (π, V ) and (ρ,W ) be representa-

tions of G. Let V =⊕r

i=1 Vi and W =⊕s

j=1Wj be decompositions of V and W into

irreducible invariant subspaces of π and ρ respectively, and let πi = π|Viand ρj = ρ|Wj

be the corresponding subrepresentations for all i and j. For any irreducible representa-

tion σ of G, let n(σ, π) be the number of subrepresentations in the above decomposition

of V that are equivalent to σ, and similarly for n(σ, ρ). Then

dim HomG(π, ρ) =∑σ

n(σ, π)n(σ, ρ),

where the sum is over all irreducible representations σ of G (up to equivalence).

Proof. By Schur’s lemma,

dim HomG(πi, ρj) =

{1 if πi ∼= ρj; and

0 otherwise.(2.1)

It is a general property of modules over a ring R that HomR(A⊕B,C) = HomR(A,C)⊕HomR(B,C) and HomR(A,B⊕C) = HomR(A,C)⊕HomR(B,C) for R-modules A, B

and C. Since representations of a group G are just modules over the group algebra CGand intertwiners are just CG-module homomorphisms, we have

HomG(π, ρ) =r⊕i=1

s⊕j=1

HomG(πi, ρj).

It now follows from (2.1) that

dim HomG(π, ρ) = dim⊕

1≤i≤r1≤j≤sπi∼=ρj

HomG(πi, ρj) =∑σ

∑1≤i≤r1≤j≤sπi∼=ρj∼=σ

1 =∑σ

n(σ, π)n(σ, ρ),

where σ runs over all irreducible representations of G (up to equivalence).

2.2. Induced representations and Mackey theory 5

Some simple consequences of this proposition are especially useful.

Corollary 2.6. Let (π, V ) be a representation of a finite groupG. If dim HomG(π, π) =

1 then π is irreducible. If dim HomG(π, π) = 2 then π is a direct sum of two inequivalent

irreducible representations.

Proof. Apply Proposition 2.5 with ρ = π.

Corollary 2.7. Let G be a finite group and let (π, V ) and (ρ,W ) be representations

of G. If dim HomG(π, ρ) = 0 then π and ρ have no equivalent nontrivial subrepresen-

tations.

Proof. If they had a nontrivial equivalent subrepresentation, they must have an equiva-

lent irreducible subrepresentation, which is impossible if dim HomG(π, ρ) = 0 by Propo-

sition 2.5.

2.2 Induced representations and Mackey theory

Definition 2.8. Let G be a group, H a subgroup of G and (π, V ) a representation

of H. Let

V G ={f : G→ V

∣∣ f(hg) = π(h)f(g) for all h ∈ H and g ∈ G}

and define the representation πG : G→ GL(V G) by(πG(k)f

)(g) = f(gk)

for all g, k ∈ G. Since h(gk) = (hg)k, we have πG(k)f ∈ V G for all f ∈ V G and k ∈ G.

Also, (πG(k)πG(l)f

)(g) =

(πG(l)f

)(gk) = πG(gkl) =

(πG(kl)f

)(g)

for all g, k, l ∈ G. So πG is indeed a representation of G on V G. We call (πG, V G) the

induced representation of G from (π, V ).

Induced representations are our main tool for creating representations of groups

from representations of their subgroups. We would often like to know about the struc-

ture of induced representations, and a good way to do this is to analyse intertwiners

between induced representations.

Proposition 2.9. Let H1 and H2 be subgroups of a finite group G. Let (π1, V1)

and (π2, V2) be representations of H1 and H2 respectively. Let

D =

{∆ : G→ HomC(V1, V2)

∣∣∣∣ ∆(h2gh1) = π2(h2) ◦∆(g) ◦ π1(h1)

for all h1 ∈ H1, h2 ∈ H2 and g ∈ G

}.

Then HomG(πG1 , πG2 ) and D are isomorphic as complex vector spaces.


Proof. For any function f : G→ V1 and any ∆ ∈ D, we can consider the convolution

∆ ∗ f : G→ V2 defined by

(∆ ∗ f)(g) =∑x∈G

∆(gx−1)f(x)

for all g ∈ G. It is easily seen from the definition of D that ∆∗f ∈ V G2 . For any ∆ ∈ D

we define the map L(∆) : V G1 → V G

2 by L(∆)(f) = ∆ ∗ f . Clearly L(∆) is C-linear,

and for all g, k ∈ G we have(πG2 (k)(∆ ∗ f)

)(g) = (∆ ∗ f)(gk) =

∑x∈G

∆(gkx−1)f(x)

=∑y∈G

∆(gy−1)f(yk) =(∆ ∗ πG1 (k)f

)(g),

so L(∆) ∈ HomG(πG1 , πG2 ).

Now we consider the map L : D → HomG(πG1 , πG2 ) given by ∆ 7→ L(∆) as defined

above. We claim this is a vector space isomorphism. It follows from the definition of

convolution that L is C-linear. Suppose first that L(∆) = 0 for some ∆ ∈ D. For any

v ∈ V1 define ev ∈ V G1 by ev(h1) = π1(h1)v for h1 ∈ H1 and ev(g) = 0 for g ∈ G \H1.

Now, since L(∆) = 0,

0 =(∆ ∗ ev

)(g) =

∑h1∈H1

∆(gh−11 )π1(h1)v = |H1|∆(g)v

for all v ∈ V1 and g ∈ G. Thus ∆ = 0, and so L is injective.

Finally, let T ∈ HomG(πG1 , πG2 ), and define ∆ : G→ HomC(V1, V2) by

∆(g)v = |H1|−1T (ev)(g)

for all g ∈ G and v ∈ V1, where ev is as defined above. ∆(g) is a C-linear map because

v 7→ ev is itself C-linear. Also,

∆(h2gh1)v = |H1|−1π2(h2)T (ev)(gh1) as T (ev) ∈ V G2

= |H1|−1π2(h2)πG2 (h1)(T (ev)

)(g)

= |H1|−1π2(h2)T(πG1 (h1)ev

)(g) as T is an intertwiner

= |H1|−1π2(h2)T(eπ1(h1)v

)(g)

= π2(h2)∆(g)π1(h1)v

for all h1 ∈ H1, h2 ∈ H2 and g ∈ G, so ∆ ∈ D. Now we can check that

(∆ ∗ f)(g) =∑x∈G

|H1|−1L(ef(x))(gx−1) = |H1|−1

∑x∈G

πG2 (x−1)(T (ef(x))

)(g)

= |H1|−1∑x∈G

T(πG1 (x−1)ef(x)

)(g) = T

(|H1|−1

∑x∈G

πG1 (x−1)ef(x)

)(g)

2.2. Induced representations and Mackey theory 7

for all f ∈ V G1 and g ∈ G. But it is easy to see that(∑

x∈G

πG1 (x−1)ef(x)

)(g) =

∑x∈G

ef(x)(gx−1) =

∑x∈H1g

π1(gx−1)f(x) = |H1|f(g)

for all f ∈ V G1 and g ∈ G, so ∆ ∗ f = T (f) for all f ∈ V G

1 . Thus L(∆) = T . Since

T ∈ HomG(πG1 , πG2 ) was arbitrary, L is surjective, and the result follows.

For ∆ ∈ D as in Proposition 2.9, the map ∆(g) determines the maps ∆(g′) for all

g′ in the double coset H1gH2. It is therefore natural to identify ∆ with its values on a

set of double coset representatives, and that is what Mackey’s theorem does.

Theorem 2.10 (Mackey’s theorem). Let H1 and H2 be subgroups of a finite group G,

and let x1, . . . , xr be a complete set of double coset representatives for H2\G/H1. Let

(π1, V1) and (π2, V2) be representations of H1 and H2, respectively. For i = 1, . . . , r, de-

fine representations (π1,i, V1) and (π2,i, V2) of Si := xiH1x−1i ∩H2 by π1,i(s) = π1(x−1

i sxi)

and π2,i(s) = π2(s). Then

HomG(πG1 , πG2 ) ∼=

r⊕i=1

HomSi(π1,i, π2,i)

as complex vector spaces.

Proof. LetW =⊕r

i=1 HomSi(π1,i, π2,i). By Proposition 2.9, it will suffice to prove that

D ∼= W . Define the linear map M : D → W by M∆ =(∆(xi)

)ri=1

. We need to check

that each ∆(xi) is an intertwiner from π1,i to π2,i, which is true because

∆(xi) ◦ π1,i(s) = ∆(xi) ◦ π1(x−1i sxi) = ∆(sxi) = π2,i(s) ◦∆(xi)

for all i and all s ∈ Si. M is clearly C-linear, and is injective since ∆(xi) determines

the values of ∆ on H1xiH2 by Proposition 2.9.

We would like to show M is surjective. Let (Ti)ri=1 ∈ W , and tentatively define

∆ : G→ HomC(V1, V2) by

∆(h2xih1) = π2(h2) ◦ Ti ◦ π1(h1)

for all i and all h1 ∈ H1, h2 ∈ H2. This gives a candidate definition of ∆(g) for all g ∈ G,

since we have a complete set of double coset representatives xi. We need to check this

definition is valid. Since the double cosets in H2\G/H1 are disjoint and have union G,

the only possible conflicting definitions occur when h2xih1 = h′2xih′1 for some i, some

h1, h′1 ∈ H1 and some h2, h

′2 ∈ H2. If that is the case, then h−1

2 h′2 = xih1(h′1)−1x−1i ∈ Si.

Using the fact that Ti ◦ π1,i

(xih1(h′1)−1x−1

i

)= π2,i

(h−1

2 h′2)◦ Ti, we have

π2(h2) ◦ Ti ◦ π1(h1) = π2(h2) ◦ π2,i(h−12 h′2) ◦ Ti ◦ π1,i(xih

′1h−11 x−1

i ) ◦ π1(h1)

= π2(h2h−12 h′2) ◦ Ti ◦ π1(h′1h

−11 h1)

= π2(h′2) ◦ Ti ◦ π1(h′1),

which shows that ∆ is well-defined. It is clear from the definition of ∆ that ∆ ∈ Dand M∆ = (Ti)

ri=1. So M is surjective, and the theorem is proved.


2.3 The Heisenberg group

In order to construct the Weil representation for SL(2, F ), we need to consider repre-

sentations of a simpler group called the Heisenberg group, which is in some respects

similar to the group of matrices of the form1 aT c

0 I b

0 0T 1

.

Definition 2.11. Suppose we are given a field F not of characteristic 2, a finite-

dimensional vector space V over F , and a nondegenerate symmetric bilinear form

B : V × V → F . The Heisenberg group is the set H = V × V × F equipped with

the multiplication

(u, v, x)(u′, v′, x′) = (u+ u′, v + v′, x+ x′ +B(u, v′)−B(v, u′)).

We need to check this is a group: multiplication is associative since

(u+ u′, v + v′, x+ x′ +B(u, v′)−B(v, u′))(u′′, v′′, x′′)

=(u+ u′ + u′′, v + v′ + v′′, x+ x′ + x′′ +B(u, v′)

+B(u, v′′) +B(u′, v′′)−B(v, u′) +B(v, u′′) +B(v′, u′′))

= (u, v, x)(u′ + u′′, v′ + v′′, x′ + x′′ +B(u′, v′′)−B(v′, u′′)).

Furthermore, (0, 0, 0) is an identity element, and (u, v, x) has an inverse (−u,−v,−x).

Remark 2.12. To find the centre of H, note that (u, v, x) is in the centre if and only if(u+u′, v+ v′, x+x′+B(u, v′)−B(v, u′)

)=(u′+u, v′+ v, x′+x+B(u′, v)−B(v′, u)

)for all (u′, v′, x′) ∈ H. This is equivalent to requiring that B(u, v′) = B(v, u′) for

all u′, v′ ∈ V . By taking v′ = 0 this implies that B(v, u′) = 0 for all u′, v′ ∈ V ,

which implies that v = 0, since B is nondegenerate. Similarly we require that u = 0.

Conversely, (0, 0, x) is in the centre of H for all x ∈ F . So the centre of H is {(0, 0, x) |x ∈ F}.

Proposition 2.13. SL(2, F ) acts on H by group automorphisms, letting(a b

c d

)· (u, v, x) = (au+ bv, cu+ dv, x)

for all(a bc d

)∈ SL(2, F ) and (u, v, x) ∈ H.

Proof. It is very straightforward to check that this is a group action. To check that

SL(2, F ) acts by automorphisms, we must check that

(au+ bv, cu+ dv, x)(au′ + bv′, cu′ + dv′, x′)

= (a(u+ u′) + b(v + v′), c(u+ u′) + d(v + v′), x+ x′ +B(u, v′)−B(v, u′)).

2.4. The structure of SL(2, F ) 9

The first two components clearly agree, so we need only check that

x+ x′ +B(au+ bv, cu′ + dv′)−B(cu+ dv, au′ + bv′) = x+ x′ +B(u, v′)−B(v, u′).

After expansion and cancellation, the left hand side reduces to

x+ x′ + (ad− bc)B(u, v′) + (bc− ad)B(v, u′),

which equals the right hand side since ad− bc = 1.

2.4 The structure of SL(2, F )

We will need a presentation for SL(2, F ) in order to lift a projective representation to a

true representation. Our presentation is inspired by the Bruhat decomposition: we only

need a single representative,(

1−1

), of the nontrivial double coset in B\SL(2, F )/B,

where B is the subgroup of SL(2, F ) whose elements are upper triangular.

Lemma 2.14. SL(2, F ) is generated by(

1 b1

)for b ∈ F ,

( a1/a

)for a ∈ F×, and

(1

−1

).

Proof. If(a bd

)∈ SL(2, F ) then d = 1/a and so(

a b

d

)=

(1 ab

1

)(a

1/a

).

If c 6= 0 then(

1−1

)(1 −a/c

1

)(a bc d

)is in SL(2, F ) and has zero as its bottom-left

entry, so it can be expressed in terms of the generators by the previous case. This gives

the decomposition(a b

c d

)=

(1 a/c

1

)(1

−1

)(1 cd

1

)(−c

−1/c

)if c 6= 0.

Before we state our presentation of SL(2, F ), we recall the formal definition of a

group given by generators and relations. Let X be a set, F the free group on X

and R a subset of F . R can be thought of as a set of words in the elements of X

and their inverses; a relation stated in the form x1 . . . xm = y1 . . . yn translates to the

word x1 . . . xmy−1n . . . y−1

1 in R. Let N be the intersection of all normal subgroups of

F containing R; this is itself a normal subgroup of F . The group generated by

X subject to the relations R is defined to be G = F/N . There is a natural

map i : X → G : x 7→ xN . The universal property of this construction states that,

if H is a group and f : X → H any map such that f(x1)ε1 . . . f(xn)εn = 1 for every

word xε11 . . . xεnn ∈ R (xi ∈ X, εi ∈ {±1}), then there exists a unique homomorphism

α : G→ H such that f = α ◦ i.

Proposition 2.15. Let F be a field. Consider the group G generated by the symbols

t(a) for a ∈ F×, n(b) for b ∈ F and w, subject to the relations

t(a1)t(a2) = t(a1a2), n(b1)n(b2) = n(b1 + b2),

t(a)n(b) = n(a2b)t(a), wt(a)w = t(−1/a),

and wn(b)w = n(−1/b)wn(−b)t(−b) if b 6= 0.

(2.2)


Then there exists a unique isomorphism α : G→ SL(2, F ) such that

t(a) 7→(a

1/a

), n(b) 7→

(1 b

1

), and w 7→

(1

−1

). (2.3)

Proof. By the universal property of a group defined by generators and relations, there

exists a unique homomorphism α satisfying (2.3) if and only if the images of the

generators in (2.3) satisfy the relations (2.2). It is easy to see that(a1

1/a1

)(a2

1/a2

)=

(a1a2

1/a1a2

),(

1 b1

1

)(1 b2

1

)=

(1 b1 + b2

1

),(

a

1/a

)(1 b

1

)=

(a ab

1/a

)=

(1 a2b

1

)(a

1/a

),

and

(1

−1

)(a

1/a

)(1

−1

)=

(−1/a

−a

).

The last relation is less obvious, but we can calculate that(1

−1

)(1 b

1

)(1

−1

)=

(−1

b −1

)=

(1/b 1

−1

)(−b 1

−1/b

)=

(1 −1/b

1

)(1

−1

)(1 −b

1

)(−b

−1/b

).

Thus there is a unique homomorphism α satisfying (2.2). By Lemma 2.14, α is surjec-

tive.

Before showing α is injective, we will make some observations about G. Since

n(0)n(0) = n(0) and t(1)t(1) = t(1), we see that n(0) = t(1) = 1, and hence t(−1)2 =

t(1) = 1. Note that the fourth relation with a = 1 gives w2 = t(−1), which shows that

wt(a) = wt(a)t(−1)2 = wt(a)w4 = t(−1/a)w3 = t(−1/a)t(−1)w = t(1/a)w

for all a ∈ F×. This will allow us to “bump” t(a) past w in our calculations.

To show α is injective, we will construct an inverse map β : SL(2, F )→ G. As we

saw in Lemma 2.14, there are two cases to consider, so we define

β

(a b

d

)= n(ab)t(a)

and β

(a b

c d

)= n(a/c)wn(cd)t(−c) if c 6= 0.

To check this is a homomorphism, we must check that

β

(A B

C D

)β

(a b

c d

)= β

(Aa+Bc Ab+Bd

Ca+Dc Cb+Dd

)(2.4)

2.4. The structure of SL(2, F ) 11

whenever AD −BC = ad− bc = 1.

If C = c = 0, we can use the third relation to “bump” t past n as follows:

n(AB)t(A)n(ab)t(a) = n(AB)n(A2ab)t(A)t(a) = n(Aa(Ab+Bd)

)t(Aa)

since ad = 1, which proves (2.4). If c = 0 but C 6= 0, checking (2.4) is just as easy:

n(A/C)wn(CD)t(−C)n(ab)t(a) = n(Aa/Ca)wn(CD)n(C2ab)t(−C)t(a)

= n(Aa/Ca)wn(Ca(Cb+Dd)

)t(−Ca)

since ad = 1. If C = 0 but c 6= 0, we have to do a little more work. Thus we can

“bump” t past n and w to get

n(AB)t(A)n(a/c)wn(cd)t(−c) = n(AB)n(A2a/c)t(A)wn(cd)t(−c)= n(A2a/c+ AB)wt(1/A)n(cd)t(−c)= n(A2a/c+ AB)wn(cd/A2)t(1/A)t(−c)= n

((Aa+Bc)/Dc

)wn(D2cd)t(−Dc)

since D = 1/A, which proves (2.4).

If C and c are both nonzero, we first calculate that

β(A BC D

)β(a bc d

)= n(A/C)wn(CD)t(−C)n(a/c)wn(cd)t(−c)= n(A/C)wn(CD)n(C2a/c)wn(cd/C2)t(c/C)

= n(A/C)wn(C(Ca+Dc)/c

)wn(cd/C2)t(c/C).

There are now two subcases, depending on whether Ca+Dc is zero or not. If Ca+Dc 6=0, then we can use the fifth relation to get rid of one w:

β(A BC D

)β(a bc d

)= n

(AC

)n( −cC(Ca+Dc)

)wn(−C(Ca+Dc)

c

)t(−C(Ca+Dc)

c

)n(cdC2

)t(cC

)= n

(A(Ca+Dc)−cC(Ca+Dc)

)wn(−C(Ca+Dc)

c

)n(d(Ca+Dc)2

c

)t(−Ca−Dc)

= n(ACa+(1+BC)c−c

C(Ca+Dc)

)wn((Ca+Dc)d(Ca+Dc)−C

c

)t(−Ca−Dc)

= n(Aa+BcCa+Dc

)wn((Ca+Dc) (1+bc)C+cdD−C

c

)t(−Ca−Dc)

= n(Aa+BcCa+Dc

)wn((Ca+Dc)(Cb+Dd)

)t(−Ca−Dc)

= β(Aa+Bc Ab+BdCa+Dc Cb+Dd

).

If Ca+Dc = 0, we observe that −c(Cb+Dd) = C(1−ad)−Dcd = C−d(Ca+Dc) = C,

and pick up from the same point we left off:

β(A BC D

)β(a bc d

)= n(A/C)w2n(cd/C2)t(c/C)

= n(A/C)n(cd/C2)t(−c/C) as w2 = t(−1)

= n(AC+cdC2

)t(− cC

)= n

(−Ac(Cb+Dd)+cd−Cc(Cb+Dd)

)t(

1Cb+Dd

)= n

(−ACb−(1+BC)d+d−C(Cb+Dd)

)t(

1Cb+Dd

)= n

((Aa+Bc)(Ab+Bd)

)t(Aa+Bc)

)= β

(Aa+Bc Ab+Bd

Cb+Dd

),


since (Aa+Bc)(Cb+Dd) = 1. We have shown β is a homomorphism.

It is now easy to see that

β(α(t(a))

)= β

( a1/a

)= n(0)t(a) = t(a),

β(α(n(b))

)= β

(1 b

1

)= n(b)t(1) = n(b),

and β(α(w)

)= β

(1

−1

)= n(0)wn(0)t(1) = w,

and hence that β ◦ α = id (since it is true on the generators of G). Therefore α is

injective and thus an isomorphism.

Chapter 3

Representations over finite fields

In this chapter, we construct all the irreducible representations of GL(2, F ) for a fi-

nite field F . We first prove the Finite Stone–von Neumann Theorem. Applied to the

Heisenberg group H, this establishes the existence of a unique irreducible represen-

tation of H with certain properties. The action described in Proposition 2.13 allows

us to construct a family of equivalent irreducible representations of H, and we then

use Schur’s lemma to show that the intertwiners among the members of this family

themselves form a projective representation of SL(2, F ). From this point on, a fair

amount of calculation is needed, first to find the projective representation of SL(2, F )

in explicit form, and then to check that the projective representation can be lifted to

a true representation of SL(2, F ), called the Weil representation.

As an interlude, we construct the principal series representations as induced rep-

resentations from the standard Borel subgroup of GL(2, F ). This provides motivation

to specialise the parameters of the Weil representation in a particular way, so that the

representation extends to a representation of GL(2, F ). We introduce cuspidality as a

defining property of the new representations, and use it to prove they are irreducible.

Once we verify all our irreducible representations are inequivalent, a simple counting

argument shows that we have found all the irreducible representations of GL(2, F ).

3.1 The Finite Stone–von Neumann Theorem

Definition 3.1. A group G is two-step nilpotent if G/Z(G) is abelian.

Throughout this section, let H be a finite two-step nilpotent group with centre Z,

and let χ0 be a character of Z. We denote the quotient space H/Z by H and the

corresponding quotient map by x 7→ x.

Proposition 3.2. There exists a bilinear, skew-symmetric pairing 〈 , 〉 : H×H → C×

satisfying 〈x, y〉 = χ0(xyx−1y−1) for all x, y ∈ H. (Bilinear and skew-symmetric have

the meanings appropriate when considering H and C× as Z-modules).

Proof. First observe that xyx−1y−1 = x y x−1y−1 = 1 as H is abelian, so xyx−1y−1 ∈ Zfor all x, y ∈ H. For any z ∈ Z we have χ0(xzy(xz)−1y−1) = χ0(xyx−1y−1zz−1) =

13

14 Chapter 3. Representations over finite fields

χ0(xyx−1y−1), and similarly we see χ0(xyzx−1(yz)−1) = χ0(xyx−1y−1), so the map

(x, y) 7→ χ0(xyx−1y−1) is well-defined. Now

〈x1x2, y〉 = χ0

(x1x2yx

−12 x−1

1 y−1)

= χ0

(x1(x2yx

−12 )x−1

1 (x2yx−12 )−1 · x2yx

−12 y−1

)=⟨x1, x2yx

−12

⟩⟨x2, y

⟩= 〈x1, y〉〈x2, y〉 as x2yx

−12 y−1 ∈ Z.

and similarly 〈x, y1y2〉 = 〈x, y1〉〈x, y2〉, so 〈 , 〉 is bilinear. Obviously 〈x, x〉 = 1, which

implies that

1 = 〈xy, xy〉 = 〈x, x〉〈x, y〉〈y, x〉〈y, y〉 = 〈x, y〉〈y, x〉

and so it is skew-symmetric.

Henceforth we assume that χ0 is generic (with respect to H), which means that

the map H → C× : x 7→ 〈x, y〉 is nontrivial for every y 6= 1 in H.

Remark 3.3. A standard result in the character theory of finite groups states that

a finite abelian group of order n has n distinct characters. Since χ0 is generic, the

characters x 7→ 〈x, y〉 are distinct, and hence these are all the characters of H.

Definition 3.4. With H and χ0 as above, a subgroup A of H is said to be isotropic

if 〈x, y〉 = 1 for all x, y ∈ A, and is said to be polarising if, for each x ∈ H, 〈x, y〉 = 1

for all y ∈ A if and only if x ∈ A.

If we define A⊥

={x ∈ H

∣∣ 〈x, y〉 = 1 for all y ∈ A}

, then A is polarising if and

only if A = A⊥

. Polarising subgroups are clearly also isotropic.

Proposition 3.5. Maximal isotropic subgroups are polarising.

Proof. Let A be a maximal isotropic subgroup of H, and take any x ∈ H such that

〈x, y〉 = 1 for all y ∈ A. Define B = 〈x〉 and consider BA, which is a subgroup of H

since H is abelian. Any element of BA can be written as xky for some k and some

y ∈ A, so the pairing applied to any two elements of BA evaluates to⟨xky, xlz

⟩= 〈x, x〉kl〈x, z〉k〈y, x〉l〈y, z〉 = 1kl1k1l1 = 1

as x ∈ A⊥. Thus BA is isotropic and contains A; by assumption we obtain BA = B.

It follows that x ∈ A, so A is polarising.

Lemma 3.6. Let F ≤ G be finite abelian groups, and let χ be a character of F . Then

there exist exactly [G : F ] characters of G which extend χ.

Proof. We proceed by induction on [G : F ]. If [G : F ] = 1, then G = F , so the only

character of G extending χ is χ itself.

3.1. The Finite Stone–von Neumann Theorem 15

Now suppose the result is true for all pairs F ≤ G with [G : F ] < n, and take a pair

F ≤ G with [G : F ] = n. Take any x ∈ G\F and let X = 〈x〉. Let k = min{i | xi ∈ F},let r be any kth root of χ(xk) in C, and tentatively define a homomorphism

ψ : FX → C× : fxi 7→ χ(f)ri.

To check that this is well-defined, note that if f1xi1 = f2x

i2 for f1, f2 ∈ F and i1, i2 ∈ Zthen xi1−i2 = f−1

1 f2 ∈ F , so i1 − i2 = mk for some m ∈ Z. It follows that χ(f−11 f2) =

χ(xi1−i2) = χ(xmk) = χ(xk)m = rkm = ri1−i2 and hence that χ(f1)ri1 = χ(x2)rx2 , so ψ

is well-defined, and it is clearly a character of FX. On the other hand, every character

of FX extending χ is of the form fxi 7→ χ(f)ψ(x)i where ψ(x)i is a kth root of χ(xk).

It follows that there are exactly k characters of FX extending χ, one for each choice

of r as a kth root of χ(xk).

Since [G : FX] < [G : F ], each character ψ of FX extending χ can be extended

to a character of G in exactly [G : FX] ways, by the inductive assumption. We also

observe that every character of G extending χ restricts to a character of FX extending

χ, so there are exactly k[G : FX] characters of G extending χ. Finally, note that

[FX : F ] = k, since every coset of F in FX is of the form Fxi, and Fxi1 = Fxi2

if and only if k | i1 − i2. We conclude that the number of extensions of χ to G is

[G : FX][FX : F ] = [G : F ].

Proposition 3.7. Let B ≤ A be isotropic subgroups of H, let A and B be their

preimages in H, and let χB be a character of B extending χ0. Then χB can be

extended to a character of A in exactly [A : B] ways. In particular, since Z is itself

isotropic, χ0 can be extended to a character of A.

Proof. Let Z0 = kerχ0. Since A is isotropic, χ0(xyx−1y−1) = 1 for all x, y ∈ A. In

particular, if a ∈ A and z ∈ Z0 then aza−1 = (aza−1z−1)z ∈ Z and χ0(aza−1) =

χ0(z) = 1, so aza−1 ∈ Z0. So Z0 is normal in A and A/Z0 is abelian.

Consider the character χ′B of B/Z0 defined by χ′B(bZ0) = χB(b) for all b ∈ B; this

is a well-defined character since Z0 ≤ kerχB. We can extend this to a character ψ of

A/Z0 in [A/Z0 : B/Z0] = [A : B] ways by Lemma 3.6 and thence derive a character

ϕ = ψ ◦ q of A, where q : A → A/Z0 is the quotient map. Since every character of A

extending χ0 must factor through q, we are done.

Lemma 3.8. Let A be a subgroup of H. Then A⊥⊥

= A.

Proof. Clearly 〈a, x〉 = 1 for all a ∈ A and x ∈ A⊥

and so A ⊆ A⊥⊥

. To prove the

reverse inclusion, let b /∈ A and let χ be the trivial character on A. Let k = min{i |bi ∈ A} and let r be any kth root of 1 except 1 itself. By the same argument as in

Lemma 3.6 there exists a character χ′ of H which is trivial on A and satisfies χ′(b) = r.

Since χ0 is generic, χ′ is given by x 7→ 〈x, y〉 for some y ∈ H. We then have y ∈ A⊥

but 〈b, y〉 = r 6= 1, so b /∈ A⊥⊥ as we wanted.


Proposition 3.9. Let A and B be polarising subgroups of H and let A and B be

their preimages in H. Let χA and χB be characters of A and B extending χ0, and let

πA and πB be the representations of H induced from χA and χB respectively. Then

dim HomH(πA, πB) = 1.

Proof. For all a ∈ A and h ∈ H, hah−1 = hah−1a−1a ∈ A, since hah−1a−1 ∈ Z ⊆ A.

In other words, A is normal in H. Similarly, B is normal in H. Hence AB is also a

normal subgroup of H.

From each double coset BxiA ∈ B\H/A choose a representative xi, and define the

character χA,i : A ∩ B → C by χA,i(s) = χA(x−1i sxi

)for all s ∈ A ∩ B. We wish to

apply Theorem 2.10 to find the dimension of HomH(πA, πB). To do so we will need to

consider the intertwiners from χA,i to χB|A∩B for each i.

Since χA,i and χB|A∩B are characters of A ∩ B, a nonzero intertwiner T from χA,ito χB|A∩B is simply a nonzero linear map C→ C such that T ◦χA,i(s) = χB(s) ◦ T for

all s ∈ A ∩ B. Since scalar transformations commute, a nonzero intertwiner exists if

and only if χA,i(s) = χB(s) for all s ∈ A∩B. Since χA,i(s)χA(s)−1 = χA(x−1i sxis

−1)

=⟨xi−1, s

⟩=⟨s, xi

⟩, a nonzero intertwiner exists if and only if

χB(s)χ−1A (s) =

⟨s, xi

⟩(3.1)

for all s ∈ A ∩B.

Since χA and χB agree on Z, χBχ−1A factorises to a character of A ∩B. This

can be extended to a character of H by Lemma 3.6, which can in turn be written

as y 7→ 〈y, xk〉 for some xk ∈ H (which we take as the representative of the kth

double coset) by Remark 3.3. For that choice of xk, (3.1) is satisfied, and hence

dim HomA∩B(χA,k, χB) = 1.

If there is some other coset AxjB with a nonzero intertwiner χHA,j → χHB , then by the

same argument (χBχ−1A )(s) = 〈s, xj〉 and hence

⟨s, xkx

−1j

⟩= 1 for all s ∈ A∩B. Thus

xkx−1j ∈ (A ∩B)⊥ = (A ∩ B)⊥. Note that if x ∈ A ∩ B then 〈x, ab〉 = 〈x, a〉〈x, b〉 = 1

for all a ∈ A and b ∈ B, so x ∈ (AB)⊥. So xkx−1j ∈ (AB)⊥⊥ = AB by Lemma 3.8,

which means that AxkB = xkAB = xjAB = AxjB. In other words, for all double

cosets AxjB other than AxkB we have dim HomA∩B(χA,j, χB) = 0.

It is now clear from Theorem 2.10 that

dim HomH(πA, πB) =∑i

dim HomA∩B(χA,i, χB) = 1.

Before we state the main theorem, we need the following definition.

Definition 3.10. If π is an irreducible representation of a finite group G and z ∈ Z(G),

then π(z)π(g) = π(g)π(z) for all g ∈ G, so π(z) is an intertwiner from π to π. By

Schur’s Lemma, π(z) is a scalar multiple, χ(z) say, of the identity map, and χ : Z(G)→C× is clearly a character. It is called the central character of G.

We restate the standing assumptions for the sake of completeness.

3.1. The Finite Stone–von Neumann Theorem 17

Theorem 3.11 (The Finite Stone–von Neumann Theorem). Let H be a finite two-

step nilpotent group with centre Z, and let χ0 be a generic character of Z. There

exists an irreducible representation π of H with central character χ0, and π is unique

up to equivalence. It may be constructed as the representation πA induced from the

character χA, where A is any polarising subgroup of H, A is its preimage in H and χAis any extension of χ0 to A.

Proof. That πA is irreducible follows from Proposition 3.9 and Corollary 2.6, taking

A = B and χA = χB. In addition, any different choice of polarising subgroup A′ ≤ H

(with preimage A′ in H) and character χA′ of A′ extending χ0 yields an irreducible

representation πA′ of H induced from χA′ , and πA′ is equivalent to πA by Proposition 3.9

and Schur’s Lemma.

Let π be any irreducible representation of H with central character χ0 and let

B ={b ∈ H

∣∣ π(b) is a scalar transformation},

observing that Z ≤ B. For b1, b2 ∈ B we have

〈b1, b2〉 = χ0(b1b2b−11 b−1

2 ) = π(b1)π(b2)π(b1)−1π(b2)−1 = 1

as these transformations are scalar, so B is isotropic. There is an obvious character

χB of B given by χB(b)v = π(b)v for all v in the space of π. By Proposition 3.5, we

can let A′ be a polarising subgroup of H containing B, with preimage A′ in H. By

Proposition 3.7, we can let χ1, . . . , χl be distinct characters of A′ extending χB, where

l = [A′ : B]. Then the induced representations πi = χHi are irreducible for all i, by

Proposition 3.9. It will suffice to prove that π is equivalent to πi, for some i.

Denote by V the space of π, and denote by Vi the space of πi for all i.

We then fix any nonzero v0 ∈ V and, for each i, define the map Ti : Vi → V by

Ti(f) =∑h∈H

f(h)π(h−1)(v0).

For all i and all g ∈ H,(Ti ◦ πi(g)

)(f) =

∑h∈H

(πi(g)f

)(h)π(h−1)(v0)

=∑h∈H

f(hg)π(h−1)(v0),

whereas (π(g) ◦ Ti

)(f) = π(g)

(∑h∈H

f(h)π(h−1)(v0)

)=∑h∈H

f(h)π(gh−1)(v0)

=∑h∈H

f(hg)π(h−1)(v0),


which shows that Ti is an intertwiner from πi to π. Since πi (for all i) and π are

irreducible, it will suffice to show that Ti0 is nonzero for some i0, as this will show π is

equivalent to πi0 and hence (by Proposition 3.9) to πi for all i.

Suppose by way of contradiction that Ti is zero for all i. Fix a set of coset repre-

sentatives a1, . . . , al for B in A′. For each i we take the function fi : H → C given

by

fi(h) =

{χi(h) if h ∈ A′

0 otherwise.

Clearly fi ∈ Vi. Now

Ti(fi) =∑a∈A′

χi(a)π(a−1)(v0),

and since

χi(ba)π((ba)−1

)(v0) = χB(b)χi(a)χB(b−1)π(a−1)(v0) = χi(a)π(a−1)(v0)

for all a ∈ A′ and b ∈ B, it follows that

0 = Ti(fi) = |B|l∑

j=1

χi(aj)π(a−1j )(v0) (3.2)

for i = 1, . . . , l.

By Schur’s orthogonality relations for characters,

δi1,i2 |A′| =∑a∈A′

χi1(a)χi2(a) (3.3)

for all i1, i2. But if a ∈ A′ and b ∈ B then

χi1(ba)χi2(ba) = χB(b)χi1(a)χB(b)χi2(a) = χi1(a)χi2(a) as |χB(b)| = 1,

which shows that the summand in (3.3) is constant on each coset of B in A′. Thus

δi1,i2|A′| = |B|l∑

j=1

χi1(aj)χi2(aj)

which shows that the vectors(χi(a1), . . . , χi(al)

)are orthogonal for i = 1, . . . , l. Thus

the matrix(χi(aj)

)li,j=1

is√

[A′ : B] times an orthogonal matrix, and is thus invertible.

Multiplying the set of equations (3.2) by the inverse matrix, we obtain π(a−1j )(v0) = 0

for all j, which is impossible as v0 6= 0.

Corollary 3.12. Let G be a group of automorphisms of H which fix the elements of

Z. Let (π, V ) be an irreducible representation of H with central character χ0. For each

g ∈ G there exists a nonzero linear map η(g) : V → V , unique up to scalar multiples,

such that

π(g · h)η(g) = η(g)π(h) (3.4)

for all h ∈ H. The maps η(g) are then invertible and their images in PGL(V ) form a

projective representation ω of G.

3.2. A projective representation of SL(2, F ) 19

Proof. Since G acts on H by automorphisms, h 7→ π(g · h) is another irreducible

representation of H on V for each g ∈ G. By Schur’s lemma, there exists an intertwiner

η(g) : V → V , unique up to scalar multiples, such that (3.4) holds, and since the two

representations are irreducible, η(g) is invertible for all g ∈ G. For all h ∈ H and

g1, g2 ∈ G we have

η(g1)η(g2)π(h) = η(g1)π(g2 · h)η(g2) = π(g1g2 · h

)η(g1)η(g2),

so by the uniqueness of η(g1g2) we have η(g1)η(g2)C× = η(g1g2)C×. This shows that

the map ω : G→ PGL(V ) with ω(g) := q(η(g)

)(where q : GL(V )→ PGL(V ) is the

quotient map) is a projective representation.

3.2 A projective representation of SL(2, F )

Let F be a finite field of odd order q, V a vector space over F of finite dimension d,

and B a nondegenerate symmetric bilinear form on V . Let H be the corresponding

Heisenberg group with centre Z = {(0, 0, x) | x ∈ F}. Let ψ be a fixed nontrivial

additive character of F , and define the character χ0 of Z by χ0(0, 0, x) = ψ(x). Let A

be the subgroup {(u, 0, x) | u ∈ V, x ∈ F}.

Proposition 3.13. H is two-step nilpotent, χ0 is generic and A is polarising.

Proof. Each left coset of Z in H looks like (u, v, x)Z = (u, v, F ) with multiplication

(u, v, F )(u′, v′, F ) = (u + u′, v + v′, F ), so H/Z is isomorphic to the direct product

V × V which is abelian. Thus H is two-step nilpotent.

Since ψ is nontrivial, there exists y ∈ F with ψ(y) 6= 1.

It is useful to calculate 〈(u, v, F ), (u′, v′, F )〉 for u, v, u′, v′ ∈ V . In fact,

〈(u, v, F ), (u′, v′, F )〉 = χ0

((u, v, 0)(u′, v′, 0)(−u,−v, 0)(−u′,−v′, 0)

)= χ0

((u+ u′, v + v′, x+ x′ +B(u, v′)−B(v, u′))

(−u− u′,−v − v′,−x− x′ +B(u, v′)−B(v, u′)))

= χ0

(0, 0, 2B(u, v′)− 2B(v, u′)

)= ψ

(2B(u, v′)− 2B(v, u′)

),

since the extra terms involving u+ u′ and v + v′ cancel out.

Suppose there is some (u, v, F ) ∈ H/Z with 〈(u, v, F ), (u′, v′, F )〉 = 1 for all

(u′, v′, F ) ∈ H/Z. Then we have ψ(2B(u, v′) − 2B(v, u′)

)= 1 for all u′, v′ ∈ V .

If u 6= 0, we can find w ∈ V with B(u,w) 6= 0 since B is nondegenerate. Then it

follows that ψ(2B(u, y

2B(u,w)w)− 2B(v, 0)

)= ψ(y) 6= 1, a contradiction. Thus u = 0

and similarly v = 0. So χ0 is generic.

Finally, for any elements (u, 0, x), (u′, 0, x′) ∈ A, we have 〈(u, 0, F ), (u′, 0, F )〉 =

ψ(2B(u, 0) − 2B(0, u′)

)= 1, so A ⊆ A

⊥. Any element not in A is of the form

(u, v, x) for v 6= 0; for that element, there exists w ∈ V with B(v, w) 6= 1, so


⟨(u, v, F

),(− y

2B(v,w)w, 0, F

)⟩= ψ(y) 6= 1, so (u, v, F ) /∈ A

⊥. Thus A = A

⊥, which

means that A is polarising.

Theorem 3.11 shows us that there is a unique irreducible representation of H with

central character χ0, and that it can be modelled as the representation of H induced

from the character χA : (u, 0, x) 7→ ψ(x) of A. That is, we consider the space Vρof functions ϕ : H → C satisfying ϕ(ah) = χA(a)ϕ(h) for a ∈ A, h ∈ H, and the

representation ρ of H on Vρ given by(ρ(g)ϕ

)(h) = ϕ(hg).

For notational convenience, we will use an equivalent representation motivated by

the observation that {(0, v, 0) | v ∈ V } is a set of right coset representatives for A in

H. Let W be the space of all functions Φ : V → C and let T : W → Vρ be the map

satisfying (TΦ)(0, v, 0) = Φ(v) for all Φ ∈ W and v ∈ V ; note that the values of TΦ

on the coset A(0, v, 0) are completely determined by (TΦ)(0, v, 0). T is easily seen to

be a vector space isomorphism, so we can define the representation π of H on W by

π(g) = T−1 ◦ ρ(g) ◦ T . Then π is by definition equivalent to ρ. We want to forget

about ρ and Vρ, so we need to calculate the action of π on V explicitly. If v ∈ V and

g = (u′, v′, x′) ∈ H then

(π(g)Φ

)(v) = T−1

(ρ(g)(TΦ)

)(v)

=(ρ(g)(TΦ)

)(0, v, 0)

= (TΦ)((0, v, 0)(u′, v′, x′)

)= (TΦ)

(u′, v + v′, x′ −B(v, u′)

)= (TΦ)

((u′, 0, x′ − 2B(v, u′)−B(u′, v′))(0, v + v′, 0)

)= ψ

(x′ −B(2v + v′, u′)

)Φ(v + v′). (3.5)

We can now use Proposition 2.13 and Corollary 3.12 to obtain a projective repre-

sentation of SL(2, F ) on W . Once again, we want to find explicit formulas for this

representation, but this time there is a certain amount of guesswork involved, because

(3.4) makes it easy to check whether η is correct but difficult to find η in the first place.

It will suffice to find the action of a set of generators of SL(2, F ).

We first consider the action of g =(

1 b1

)on W . By (3.5),

(π(u′, v′, x′)Φ

)(v) = ψ

(x′ −B(2v + v′, u′)

)Φ(v + v′),

and(π(u′ + bv′, v′, x′)Φ

)(v) = ψ

(x′ −B(2v + v′, u′ + bv′)

)Φ(v + v′)

= ψ(x′ −B(2v + v′, u′)

)· ψ(bB(v, v)− bB(v + v′, v + v′)

)Φ(v + v′).

(3.6)

This last equation strongly suggests an intertwiner from π(u′, v′, x′) to π(u′+ bv′, v′, x′)

must involve some factor like ψ(bB(v, v)

). If we make the guess

(η(

1 b1

)Φ)(v) =

ψ(bB(v, v))Φ(v) for v ∈ V , then the calculation (3.6) shows that (3.4) holds.


Now we consider g =(aa−1

). By (3.5),(

π(u′, v′, x′)Φ)(v) = ψ

(x′ −B(2v + v′, u′)

)Φ(v + v′),

and(π(au′, a−1v′, x′)Φ

)(v) = ψ

(x′ −B(2v + a−1v′, au′)

)Φ(v + a−1v′)

= ψ(x′ − 2B(av, u′)−B(v′, u′)

)Φ(v + a−1v′).

The term −2B(av, u′) suggests an intertwiner involving the replacement of v with av.

If we make the guess(η(aa−1

)Φ)(v) = LaΦ(av) for v ∈ V , then (3.4) holds. (We

include the arbitrary constant La 6= 0, to be chosen later, to make it easier to lift to a

true representation.)

Finally we consider g =(

1−1

). By (3.5),(

π(u′, v′, x′)Φ)(v) = ψ

(x′ − 2B(v, u′)−B(v′, u′)

)Φ(v + v′),

and(π(v′,−u′, x′)Φ

)(v) = ψ

(x′ − 2B(v, v′) +B(u′, v′)

)Φ(v − u′).

Comparison of the terms within the ψ-factor is fruitless this time. However, we have

the luxury of V being finite, so we consider the most general linear transformation and

deduce the correct intertwiner. Let(η

(0 1

−1 0

)Φ

)(v) =

∑u∈V

f(v, u)Φ(u)

for some coefficients f(v, u) ∈ C. We want (3.4) to hold, so we compute, remembering

that g =(

1−1

)and h = (u′, v′, x′):(

π(g · h)(η(g)Φ

))(v) = ψ

(x′ − 2B(v, v′) +B(v′, u′)

)(η(g)Φ

)(v − u′)

= ψ(x′ − 2B(v, v′) +B(v′, u′)

)∑u∈V

f(v − u′, u)Φ(u)

=∑u∈V

ψ(x′ − 2B(v, v′) +B(v′, u′)

)f(v − u′, u)Φ(u),

whereas(η(g)

(π(h)Φ

))(v) =

∑u∈V

f(v, u)(π(u′, v′, x′)Φ

)(u)

=∑u∈V

f(v, u)ψ(x′ − 2B(u, u′)−B(u′, v′)

)Φ(u+ v′)

=∑u∈V

f(v, u− v′)ψ(x′ − 2B(u− v′, u′)−B(u′, v′)

)Φ(u)

=∑u∈V

f(v, u− v′)ψ(x′ − 2B(u, u′) +B(u′, v′)

)Φ(u).

Thus it will suffice that

ψ(−2B(v, v′))f(v − u′, u) = ψ(−2B(u, u′))f(v, u− v′)


for all u, v, u′, v′ ∈ V , and the obvious choice satisfying this equation is f(u, v) :=

ψ(2B(u, v)). For Φ ∈ W we define the Fourier transform Φ of Φ by

Φ(v) =∑u∈V

ψ(2B(u, v))Φ(u)

for all v ∈ V . Then the map η(

1−1

): W → W defined by η

(1

−1

)Φ = KΦ satisfies

(3.4), where K 6= 0 is a constant yet to be determined.

3.3 A true representation of SL(2, F )

By Corollary 3.12, we have now found a projective representation ω of SL(2, F ) on V .

We wish to lift this to a true representation of SL(2, F ). It will be enough to ver-

ify that η preserves the relations in Proposition 2.15, as that will show η is a true

representation.

For neatness we abbreviate our notation by writing t(a) · Φ and n(b) · Φ instead of

η(α(t(a))

)Φ and η

(α(n(b))

)Φ, and so on; we need to check this dot action satisfies the

relations. The first three relations are easily checked, if we impose the condition that

La1La2 = La1a2 for a1, a2 ∈ F×:(t(a1) · t(a2) · Φ

)(v) = La1

(t(a2) · Φ

)(a1v)

= La1La2Φ(a2a1v) =(t(a1a2) · Φ

)(v);(

n(b1) · n(b2) · Φ)(v) = ψ

(b1B(v, v)

)(n(b2) · Φ

)(v)

= ψ((b1 + b2)B(v, v)

)Φ(v) =

(n(b1 + b2) · Φ

)(v);

and(t(a) · n(b) · Φ

)(v) = La

(n(b) · Φ

)(av) = Laψ

(bB(av, av)

)Φ(av)

= ψ(a2bB(v, v)

)(t(a) · Φ

)(v) =

(n(a2b) · t(a) · Φ

)(v).

To verify the fourth relation, we calculate:(w · t(a) · w · Φ

)(v) = K

∑u∈V

ψ(2B(u, v)

)(t(a) · w · Φ

)(v)

= K∑u∈V

ψ(2B(u, v)

)La

(K∑w∈V

ψ(2B(w, au)

)Φ(w)

)= LaK

2∑u,w∈V

ψ(2B(u, v + aw)

)Φ(w)

= LaK2∑w∈V

(∑u∈V

ψ(2B(u, v + aw)

))Φ(w).

We evaluate this using a standard character trick. Since ψ is nontrivial, there is some

f ∈ F with ψ(f) 6= 1. Since B is nondegenerate, if v+aw 6= 0 there exists some u0 ∈ Vwith b0 := B(u0, v+aw) 6= 0. If we let u1 := u0f/2b0 then ψ

(2B(u1, v+aw)

)= ψ(f) 6=

1, and so∑u∈V

ψ(2B(u, v + aw)

)=∑u∈V

ψ(2B(u+ u1, v + aw)

)= ψ(f)

∑u∈V

ψ(2B(u, v + aw)

),

3.3. A true representation of SL(2, F ) 23

which shows that the sum is zero. On the other hand, if v + aw = 0 the above sum is

obviously |V | = qd. Thus(w · t(a) · w · Φ

)(v) = LaK

2qdΦ(−v/a) = LaL−1−1/aK

2qd(t(−1/a) · Φ

)(v).

But we have already had to insist that L• is a character on F×, so LaL−1−1/a = L−a2 =

L−1La2 . We want the fourth relation to be preserved regardless of the value of a, so we

need La2 to be independent of a. Since L• is a character, this means we need La2 = 1

for all a. This means that L• is either the quadratic character ξ on F× or the trivial

character. (Recall that F× is cyclic and of even order q − 1, so the squares form a

subgroup of index two, and the quadratic character sends the squares to 1 and the

nonsquares to −1.) It then follows, if we want the fourth relation to be preserved, that

we need L−1K2qd = 1.

We now check the fifth and final relation, bearing in mind the constraints on K

and L• already determined. We calculate:(n(−1/b) · w · n(−b) · t(−b) · Φ

)(v)

= ψ(−B(v, v)/b

)(w · n(−b) · t(−b) · Φ

)(v)

= Kψ(−B(v, v)/b

)∑u∈V

ψ(2B(u, v)

)(n(−b) · t(−b) · Φ

)(u)

= KL−b∑u∈V

ψ(−B(v, v)/b+ 2B(u, v)− bB(u, u)

)Φ(−bu)

= KL−b∑u∈V

ψ(−B(v − bu, v − bu)/b

)Φ(−bu)

= KL−b∑u∈V

ψ(−B(u+ v, u+ v)

/b)Φ(u),

whereas (w · n(b) · w · Φ

)(v)

= K∑u∈V

ψ(2B(u, v)

)(n(b) · w · Φ

)(u)

= K∑u∈V

ψ(2B(u, v) + bB(u, u)

)(w · Φ

)(u)

= K2∑u,w∈V

ψ(2B(u, v) + bB(u, u) + 2B(w, u)

)Φ(w)

= K2∑u,w∈V

(ψ(bB(u+ (v + w)/b, u+ (v + w)/b

))· ψ(−B(v + w, v + w)/b

)Φ(w)

)

= K2∑w∈V

(∑u∈V

ψ(bB(u, u)

))ψ(−B(v + w, v + w)/b

)Φ(w),

so the relation is preserved if K∑

u∈V ψ(bB(u, u)

)= L−b for all b ∈ F×. We need to

evaluate the sum in this expression.


Lemma 3.14 (Diagonalisation of symmetric bilinear forms). Let F be a field whose

characteristic is not 2. Let B be a nondegenerate symmetric bilinear form over a finite-

dimensional F -vector space V . Then there exists a basis b1, . . . , bk of V such that

B(bi, bj) = 0 if i 6= j and, for each i, B(bi, bi) 6= 0.

Proof. We proceed by induction on dimV . When dimV = 0, the statement is vac-

uously true (the empty basis works). Now assume the statement for all F -vector

spaces of dimension less than d, and take V of dimension d. Since B is nonzero and

B(u, v) = 14

(B(u+v, u+v)−B(u−v, u−v)

), there is some b1 ∈ V with B(b1, b1) 6= 0.

Let V ′ = (Fb1)⊥; then V = Fb1 ⊕ V ′, so dimV ′ = d− 1 and B|V ′×V ′ is nondegenerate

(since if there is v ∈ V ′ with B(v, u) = 0 for all u ∈ V ′, then also B(v, b1) = 0 and so

B(v, u) = 0 for all u ∈ V , so v = 0). The inductive assumption then implies there is a

basis b2, . . . , bd of the complement V ′ so that b1, . . . , bd is a suitable basis of V .

Let b1, . . . , bd be a basis of V such that B(bi, bj) = fiδij for all i, j, where fi ∈ F×.

Then ∑u∈V

ψ(bB(u, u)

)=

∑x1,...,xd∈F

ψ(b(x2

1f1 + . . .+ x2dfd)

)=

∑x1,...,xd∈F

ψ(bf1x21) . . . ψ(bfdx

2d)

=d∏i=1

∑x∈F

ψ(bfix2). (3.7)

We want to evaluate these sums.

Lemma 3.15. There exists ε0 ∈ {±1,±i} such that ε20 = ξ(−1) and∑

x∈F

ψ(tx2) = ε0 ξ(t)√q

for all t ∈ F×.

Proof. First observe that y ∈ F× can be written as a square in exactly two ways if

ξ(y) = 1 and in exactly no ways if ξ(y) = −1, and 0 can be written as a square in

exactly one way. Thus∑x∈F

ψ(tx2) = 1 +∑y∈F×

(1 + ξ(y)

)ψ(ty) =

∑y∈F

ψ(ty) +∑y∈F×

ξ(y)ψ(ty).

But∑

y∈F ψ(ty) = 0 by the same character trick as before, since t 6= 0. We calculate


the modulus of the second sum:∣∣∣∣∣∑y∈F×

ξ(y)ψ(ty)

∣∣∣∣∣ =∑

y,z∈F×ξ(y)ψ(ty)ξ(z)ψ(tz)

=∑

y,z∈F×ξ(y/z)ψ

(t(y − z)

)=∑

r,z∈F×ξ(r)ψ

(tz(r − 1)

)(letting r = y/z)

=∑r∈F×

(∑z∈F×

ψ(tz(r − 1)

))ξ(r).

A third application of the character trick yields

∑x∈F×

ψ(tx(r − 1)

)=∑x∈F

ψ(tx(r − 1)

)− 1 =

{q − 1 if r = 1;

−1 otherwise.

Thus ∣∣∣∣∣∑y∈F×

ξ(y)ψ(ty)

∣∣∣∣∣ = (q − 1)ξ(1) +∑

r∈F×\{1}

(−1)ξ(r)

= q −∑r∈F×

ξ(r) = q

by the character trick, applied this time to ξ. But∑y∈F×

ξ(y)ψ(ty) =∑y∈F×

ξ(y)ψ(−ty) as ξ(y) ∈ {±1}

=∑y∈F×

ξ(−y)ψ(ty)

= ξ(−1)∑y∈F×

ξ(y)ψ(ty),

so the sum is real if ξ(−1) = 1 and purely imaginary if ξ(−1) = −1. Pick ε0 ∈ {±1,±i}such that

∑y∈F× ξ(y)ψ(y) = ε0

√q. Then ε2

0 = ξ(−1) and∑y∈F×

ξ(y)ψ(ty) =∑y∈F×

ξ(y/t)ψ(y) = ξ(t)∑y∈F×

ξ(y)ψ(y) = ε0 ξ(t)√q

for all t ∈ F×. This proves the lemma.

Each factor in the product (3.7) can therefore be written as ε0 ξ(bfi)√q, so we have

∑u∈V

ψ(bB(u, u)

)=

d∏i=1

ε0 ξ(bfi)√q = εξ(b)dqd/2


where ε = εd0 ξ(f1 . . . fd).

Returning to the relations that need to be checked, our previous calculations show

the relations will be preserved if

L−1K2qd = 1 (3.8)

and Kεξ(b)dqd/2 = L−b for all b ∈ F×. (3.9)

Recall that L• is either the trivial or quadratic character on F×, and observe that the

only dependence of the left hand side of (3.9) on b is in the factor ξ(b)d. Thus, in order

for the relations to hold, Lbξ(b)−d must be a character of F× and independent of b, so

Lb = ξ(b)d for all b ∈ F×. Then (3.9) gives

K = ε−1L−1q−d/2 = ε−1ξ(−1)dq−d/2,

and fortunately then

ξ(−1)dK2qd = ξ(−1)dε−2q−dqd = 1

as ε2 = ε2d0 = ξ(−1)d, so (3.8) holds as well. In other words, η is a true representation

of SL(2, F ), called the Weil representation for SL(2, F ), which is given by the

following formulas: (η

(1 b

1

)Φ

)(v) = ψ

(bB(v, v)

)Φ(v) for b ∈ F ,(

η

(a

1/a

)Φ

)(v) = ξ(a)d Φ(av) for a ∈ F×,

and

(η

(1

−1

)Φ

)(v) = δq−d/2

∑u∈V

ψ(2B(u, v)

)Φ(u),

where δ = ε−1ξ(−1)d.

3.4 A motivating example

Before embarking on the general case, we remark on an easily constructed class of rep-

resentations which help to motivate the general construction. Let T be the subgroup

of GL(2, F ) consisting of the diagonal matrices, and let B be the so-called Borel sub-

group of GL(2, F ), consisting of the upper triangular matrices. Incidentally, T stands

for torus, which means an algebraic group defined over F which is isomorphic over some

extension of F to a direct product of copies of F× (by analogy with the geometric torus

T× T). Tori provide an important way to parametrise the irreducible representations

of GL(2, F ) (see [1], p. 401), but we will not examine this aspect further.

Let χ1 and χ2 be characters of F× and let χ : T → C× :(a1

a2

)7→ χ1(a1)χ2(a2)

be the corresponding character of T . Clearly χ extends to a character of B (which we

will also call χ) by

χ

(a b

d

)= χ1(a)χ2(d). (3.10)

3.4. A motivating example 27

We let B(χ1, χ2) be the representation of GL(2, F ) induced from the character χ of B.

Each such representation has dimension [GL(2, F ) : B] = (q2 − 1)(q2 − q)/q(q − 1)2 =

q + 1. Mackey theory again comes in helpful to compute the dimensions of the spaces

of intertwiners between these representations.

Lemma 3.16. Let χ1, χ2, µ1 and µ2 be characters of F×. Then

dim HomGL(2,F )

(B(χ1, χ2),B(µ1, µ2)

)=

0 if {χ1, χ2} 6= {µ1, µ2};1 if {χ1, χ2} = {µ1, µ2}, χ1 6= χ2;

2 if χ1 = χ2 = µ1 = µ2.

Proof. The Bruhat decomposition for GL(2, F ) states that GL(2, F ) = B t B(

11

)B;

this is true because(

11

)(1 −a/c

1

)(a bc d

)∈ B if c 6= 0. It will suffice to calculate the

dimension of D as defined in Proposition 2.9.

Let χ and µ be the characters of B obtained by (3.10) from (χ1, χ2) and (µ1, µ2)

respectively.

Take any ∆ ∈ D. Then ∆ takes its values in HomC(C,C), which we may identify

with C via ϕ↔ ϕ(1). Furthermore, ∆ is determined by its values on the double coset

representatives(

11

)and

(1

1

), so certainly dimD ≤ 2. By definition of D,

∆(

11

)= µ1(x)µ2(y)∆

(1

1

)χ1(x)−1χ2(y)−1

and ∆(

11

)= µ1(x)µ2(y)∆

(1

1

)χ1(y)−1χ2(x)−1

for all x, y ∈ F×. If χ1 6= µ1, there is some x ∈ F× such that χ1(x) 6= µ1(x), which

shows that ∆(

11

)= 0. Similarly, if χ2 6= µ2 then ∆

(1

1

)= 0, and if χ1 6= µ2 or

χ2 6= µ1 then ∆(

11

)= 0. In other words, dimD = 0 unless {χ1, χ2} = {µ1, µ2}, and

dimD ≤ 1 unless χ1 = χ2 = µ1 = µ2.

We now show the bounds above are met. If (χ1, χ2) = (µ1, µ2) then the function

∆1 : G→ C given by

∆1(g) =

{χ(g) if g ∈ B,

0 otherwise,

is easily seen to be in D. If (χ1, χ2) = (µ2, µ1) then we tentatively define ∆2 : G→ Cby

∆2(g) =

{χ(b)µ(b′) if g = b

(1

1

)b′ for b, b′ ∈ B,

0 otherwise.

This is well-defined because χ(b)µ(b′) = 1 if b(

11

)b′ =

(1

1

)for b, b′ ∈ B, and ∆2 is

then clearly an element of D. We have exhibited a nonzero element of D in the case

{χ1, χ2} = {µ1, µ2}, χ1 6= χ2, and two linearly independent elements of D in the case

χ1 = χ2 = µ1 = µ2, so the lemma is proved.

Proposition 3.17. The representations B(χ1, χ2) for χ1 6= χ2 are irreducible and

pairwise inequivalent, except that B(χ1, χ2) ∼= B(χ2, χ1). (They are known as the

principal series representations.)


Proof. This follows from Lemma 3.16 and Corollaries 2.6 and 2.7.

Proposition 3.18. Each representation B(χ1, χ1) is the direct sum of the character

χ1 : g 7→ χ1(det g) of G with an irreducible representation of dimension q (known as a

special representation). These representations are all pairwise inequivalent.

Proof. We observe that B(χ1, χ1) has a subrepresentation equivalent to the given char-

acter. To be specific, the function f : GL(2, F )→ C given by f(g) = χ1(g) = χ1(det g)

satisfies f(bg) = χ1(det b)χ1(det g) = χ(b)χ1(det g) for all b ∈ B and g ∈ GL(2, F ),

where χ is the character of B obtained from (χ1, χ1) by (3.10). So f lies in the space

of B(χ1, χ1), and Cf is clearly an invariant subspace equivalent to χ1. It follows from

Lemma 3.16 and Corollary 2.6 that Cf has a q-dimensional complement which is ir-

reducible. It then follows from Corollary 2.7 that χ1 and the special representation

corresponding to χ1 are inequivalent across choices of χ1.

3.5 Extending the representation to GL(2, F )

We would like to obtain a representation of GL(2, F ) from η. It turns out that the

paradigm of section 3.4 is extended in a surprising way: in that section we considered

multiplicative characters of the two-dimensional F -algebra of diagonal matrices (known

as the split case); in general we may also consider characters of the quadratic field

extension of F (which is unique up to isomorphism). Let E be either of these two F -

algebras. In the split case, rather than dealing with diagonal matrices, we will consider

the F -algebra E = F ⊕ F , which is isomorphic but more convenient to write.

The parameters V andB of section 3.2 need to be specialised. First denote by x 7→ x

a nontrivial F -algebra automorphism of E; specifically, we will take (a, b) 7→ (b, a) in

the split case and the (unique) nontrivial Galois automorphism x 7→ xq in the case of

a field extension. The trace and norm maps on E are then given by tr : x 7→ x+ x

and N : x 7→ xx respectively, regarding the images as elements of F (via (x, x) 7→ x

in the split case). We then take V = E (so d = 2 and all factors ξ(·)d disappear) and

B(u, v) = 12

tr(uv), which is clearly a nondegenerate symmetric bilinear form on V .

Note that then B(u, u) = 12(uu + uu) = N(u). Let E×1 be the set of elements of E of

norm 1. The following lemma is very important:

Lemma 3.19. The norm map N : E → F is surjective.

Proof. In the split case, N(a1, a2) = a1a2, so N(a1, 1) = a1 for all a1 ∈ F , so N is

surjective. Otherwise, E is a field extension of F . Zero is clearly a norm, as N(0) =

0 0 = 0. For the rest, we know N(x) = xq+1 for x ∈ E×, so N : E× → F× is a group

homomorphism. Moreover, the equation xq+1 = 1 can have at most q + 1 roots, since

E is a field, so | kerN | ≤ q + 1. Thus |N(E×)| = [E× : kerN ] ≥ (q2 − 1)/(q + 1) =

q − 1 = |F×|, so N is surjective.

3.5. Extending the representation to GL(2, F ) 29

We need to find a way to extend the representation η of SL(2, F ) to GL(2, F ).

We have not yet used any characters of E×, and the representation space of η has

dimension q2, which is rather large. Fix a character χ of E× and consider the subspace

W (χ) ={

Φ : E → C∣∣ Φ(tx) = χ(t)−1Φ(x) for all t ∈ E×1

}.

This subspace is invariant under the action of η. To see this, we take Φ ∈ W (χ) and

t ∈ E×1 , and check each of the three families of generators in turn (where, as usual,

a ∈ F× and b ∈ F ):

ψ(bN(tv)

)Φ(tv) = ψ

(bN(t)N(v)

)χ(t)−1Φ(v) = χ(t)−1ψ

(bN(v)

)Φ(v);

Φ(a(tv)

)= Φ(tav) = χ(t)−1Φ(av);

and ∑u∈V

ψ(tr(utv)

)Φ(u) =

∑u∈V

ψ(tr(ut−1v)

)Φ(u)

=∑u∈V

ψ(tr(uv)

)Φ(tu) = χ(t)−1

∑u∈V

ψ(tr(uv)

)Φ(u).

We want to extend this representation to all of GL(2, F ). To do so, we need to

understand in what circumstances a representation of a subgroup can be extended

to a representation of the whole group. The following lemma gives an extension in

circumstances that satisfy our needs:

Lemma 3.20. Let G be a group. Let H be a subgroup and M a normal subgroup of

G such that MH = G and M ∩H = 1. Then G = M oH. Suppose in addition that σ

is a representation of M and τ a representation of H, both on the space V , and that

τ(h)σ(m)τ(h)−1 = σ(hmh−1)

for all m ∈M and h ∈ H. Then the map

π : G→ GL(V ) : mh 7→ σ(m)τ(h)

is a representation of G.

Proof. If m1h1 = m2h2 for m1,m2 ∈ M and h1, h2 ∈ H, then h1h−12 = m−1

1 m2, which

equals 1 since M∩H = 1. So every element of G can be written uniquely as mh for m ∈M , h ∈ H. Then the multiplication is given by m1h1m2h2 = m1(h1m2h

−11 )h1h2, which

shows that G is the semidirect product M oH, where H acts on M by conjugation.

To check π is a representation, we need only check that it is a homomorphism, in

other words that

σ(m1)τ(h1)σ(m2)τ(h2) = σ(m1(h1m2h

−11 ))τ(h1h2)

for all m1,m2 ∈ M and h1, h2 ∈ H. Since σ and τ are homomorphisms, this is

equivalent to

τ(h1)σ(m2)τ(h1)−1 = σ(h1m2h−11 ),

which is what we have assumed.


Let G = GL(2, F ), M = SL(2, F ) and H ={(

a1

) ∣∣ a ∈ F×}

in Lemma 3.20.

Obviously M EG, M ∩H = 1 and MH = G by the simple decomposition(a b

c d

)=

(a

ad−bc bc

ad−bc d

)(ad− bc

1

).

Thus G = M oH, and if we can find compatible representations on G and H we will

have a representation of the whole of G. We have already found a representation of

M : for all g ∈ SL(2, F ), denote by σ(g) the restriction of η(g) to W (χ), where for

neatness we suppress from our notation the dependence of σ on the character χ. Then

σ is a representation of M on W (χ).

It is less obvious what representation τ of H we should consider, but it turns out

that the following choice works. We define(τ

(a

1

)Φ

)(v) = χ(c)Φ(cx)

for any c ∈ E× with N(c) = a. Such a c exists by Lemma 3.19. We need to check

that the right hand side is well-defined. Indeed, if N(c) = N(c′) = a then c/c′ ∈E×1 , so χ(c)−1Φ(cx) = χ(c)χ(c/c′)−1Φ(c′x) = χ(c′)Φ(c′x), so the right-hand side is

well-defined. If Φ ∈ W (χ) then χ(c′)Φ(c′tx) = χ(t)−1χ(c′)Φ(c′x) for t ∈ E×1 , so

θ(χ)(a

1

)Φ ∈ W (χ), which shows τ

(a

1

)maps W (χ) into itself. Finally, if a, a′ ∈ F×

and c, c′ ∈ E× with N(c) = a and N(c′) = a′, then N(cc′) = aa′; thus(τ(a

1

)τ(a′

1

)Φ)

(x) = χ(c′)χ(c)Φ(cc′x) = χ(cc′)Φ(cc′x) =(τ(aa′

1

)Φ)

(x),

so τ is in fact a representation of H on W (χ).

By Lemma 3.20, all that remains to be checked is that

τ

(a

1

)σ(n) τ

(1/a

1

)= σ

((a

1

)n

(1/a

1

))for all n ∈ SL(2, F ), and it will suffice to check this just in the case where n is one

of our generators of SL(2, F ), since the conjugate of a product is the product of the

conjugates. We will again denote the actions of τ and σ by a centred dot, and we

introduce the notation s(a) for the matrix(a

1

)in addition to the the notations from

Proposition 2.15 for generators of SL(2, F ). Then we calculate that(a

1

)(1 b

1

)(1/a

1

)=

(1 ab

1

),(

a

1

)(A

1/A

)(1/a

1

)=

(A

1/A

)and

(a

1

)(1

−1

)(1/a

1

)=

(a

−1/a

)=

(a

1/a

)(1

−1

),

3.6. Cuspidality and irreducibility 31

whereas, if N(c) = a:(s(a) · n(b) · s(1/a) · Φ

)(v) = χ(c)ψ

(bN(cv)

)(s(1/a) · Φ

)(cv)

= χ(c)ψ(abN(v)

)χ(1/c)Φ(v) =

(n(ab) · Φ

)(v);(

s(a) · t(A) · s(1/a) · Φ)(v) = χ(c)χ(1/c)Φ(cAc−1v) =

(t(A) · Φ

)(v); and(

s(a) · w · s(1/a) · Φ)(v) = χ(c) δq−1

∑u∈E

ψ(tr(ucv)

)χ(1/c)Φ(u/c)

= δq−1∑u∈E

ψ(tr(uav)

)Φ(u) =

(t(a) · w · Φ

)(v),

where in the last line we replaced u with cu in the summation. We have now shown

the map θ(χ) : GL(2, F )→ GL(W (χ)

)defined by

θ(χ)

(a b

c d

)= σ

(a

ad−bc bc

ad−bc d

)τ

(ad− bc

1

)is a representation. To be explicit, this representation is given on generators ofGL(2, F )

by (θ(χ)

(1 b

1

)Φ

)(v) = ψ

(bN(v)

)Φ(v) for b ∈ F ,(

θ(χ)

(a

1/a

)Φ

)(v) = Φ(av) for a ∈ F×,(

θ(χ)

(a

1

)Φ

)(v) = χ(c)Φ(cv) for a ∈ F×, where N(c) = a,

and

(θ(χ)

(1

−1

)Φ

)(v) = δq−1

∑u∈E

ψ(tr(uv)

)Φ(u),

for all Φ ∈ W (χ) and all v ∈ E. Here δ = ε−20 ξ−1(f1f2) = ξ(−f1f2), where f1 and

f2 are constants obtained from diagonalising N according to Lemma 3.14. In the

split case, N(a, b) = ab =(a+b

2

)2 −(a−b

2

)2, so δ = ξ

(−1(−1)

)= 1. In the field

extension case, E = F (√d) for some nonsquare d ∈ F× and N

(a + b

√d)

= a2 − b2d,

so δ = ξ(−1(−d)

)= ξ(d) = −1.

Remark 3.21. Throughout the last two sections, we have assumed F was of odd order.

This was necessary in order to use Lemma 3.14 to diagonalise a bilinear form, among

other results. However, it turns out that the representation defined by the equations

above is a true representation of GL(2, F ) even when F is of even order. This is easily

checked by verifying that the representing transformations satisfy the relations required

by Proposition 2.15 and Lemma 3.20. We will omit these calculations.

3.6 Cuspidality and irreducibility

In the split case, it transpires that the representation we have found is equivalent to a

principal series representation.


Proposition 3.22. Let E = F ⊕F and let χ : E× → C× be a multiplicative character

of E which is nontrivial on E×1 . Construct the representation(θ(χ),W (χ)

)as in

section 3.5. Let χ1 and χ2 be the multiplicative characters of F such that χ(a, b) =

χ1(a)χ2(b) for all a, b ∈ F×. Then W (χ) ∼= B(χ1, χ2).

Proof. It is easy to see that E×1 = {(t, 1/t) | t ∈ F×}. Since χ(E×1 ) 6= 1, we know that

χ1(x)χ2(1/x) 6= 1 for some x ∈ F×, so χ1 6= χ2. Proposition 3.17 then tells us that

B(χ1, χ2) is irreducible.

Observe that dimB(χ1, χ2) = [GL(2, F ) : B] = q + 1. To find the dimension of

W (χ), observe that the values of Φ ∈ W (χ) on E× are determined by the values

on the coset representatives of E×1 in E×. Furthermore, Φ(0, 0) = χ(t, 1/t)Φ(0, 0)

for all t ∈ E×1 , which means that Φ(0, 0) = 0 as χ is nontrivial on E×1 . Finally,

Φ(0, t) and Φ(t, 0) for t ∈ F× are determined by Φ(0, 1) and Φ(1, 0) respectively. Thus

dimW (χ) = [E× : E×1 ] + 2 = (q − 1)2/(q − 1) + 2 = q + 1.

It will suffice to exhibit a nonzero intertwiner T : W (χ)→ B(χ1, χ2), as then W (χ)

must have an irreducible invariant subspace equivalent to B(χ1, χ2) by Proposition 2.5,

and that subspace must be the whole space W (χ) since dimW (χ) = dimB(χ1, χ2).

Define T by

(TΦ)(g) =(θ(χ)(g)Φ

)(0, 1)

for Φ ∈ W (χ). To check that TΦ ∈ B(χ1, χ2), we need to check that (TΦ)(bg) =

χ(b) (TΦ)(g) for all g ∈ GL(2, F ) and b ∈ B. By taking Ψ = θ(χ)(g)Φ, it suffices to

check that(θ(χ)(b)Ψ

)(0, 1) = χ(b) Ψ(0, 1) for all b ∈ B and Ψ ∈ W (χ). Indeed,(

θ(χ)

(a b

d

)Ψ

)(0, 1) =

(σ

(1/d

d

)σ

(1 db

1

)τ

(ad

1

)Ψ

)(0, 1)

= χ(ad, 1)Ψ(0, 1/d)

= χ(a, d)Ψ(0, 1)

for all(a bd

)∈ B and Ψ ∈ W (χ). It is clear from the definition of T that(

T θ(χ)(h)Φ)(g) = (TΦ)(gh) =

(π(h)TΦ

)(g)

for all Φ ∈ W (χ) and g, h ∈ H (where π denotes the representation B(χ1, χ2)), so T

intertwines θ(χ) and B(χ1, χ2). Finally, T is nonzero because (TΦ)(1) = Φ(0, 1) 6= 0

for some Φ ∈ W (χ), so we are done.

In the case where E is a field, θ(χ) is a new representation not seen before. To

analyse it, we consider the subgroup U of GL(2, F ) defined by

U =

{(1 b

1

) ∣∣∣∣ b ∈ F}.We can get useful information about a representation π of GL(2, F ) by considering the

restriction of π to U . If F is considered as an additive group, then U ∼= F via the map(1 b

1

)7→ b. Thus the irreducible representations of U are in bijection with the additive

characters of F . We need the following important lemma:

3.6. Cuspidality and irreducibility 33

Lemma 3.23. If ψ is a nontrivial additive character of F , then every additive character

of F is of the form x 7→ ψ(ax) for some a ∈ F , and all these characters are distinct.

Proof. Certainly x 7→ ψ(ax) is an additive character of F for a ∈ F . We know there

is some c with ψ(c) 6= 1. If a 6= b then for x = c/(a − b) we have ψ((a − b)x

)6= 1, so

ψ(ax) 6= ψ(bx). Thus the characters x 7→ ψ(ax) for a ∈ F are all distinct. Since there

are exactly q additive characters of F , they must all be of this form.

By Maschke’s theorem, any representation (π, V ) of GL(2, F ), when considered as a

representation of U , breaks down as a direct sum of irreducible U -invariant subspaces,

each of which is one-dimensional since U is abelian. On each of these subspaces there is

some a ∈ F such that the action of U on that subspace is given by π(

1 b1

)(v) = ψ(ab)v.

For any a ∈ F , the a-isotypic subspace is defined as

Va =

{v ∈ V

∣∣∣∣ π(1 b

1

)(v) = ψ(ab)v for all b ∈ F

},

which is the union of some of the U -invariant subspaces previously mentioned; it follows

that

V =⊕a∈F

Va.

However, the extra structure obtained from π being a representation of the whole

of GL(2, F ) makes for an interesting action on the set of isotypic subspaces of V . To

be precise, pick any r ∈ F×. If a ∈ F and v ∈ V is such that π(

1 b1

)(v) = ψ(ab)v for

all b ∈ B, then

π

(1 b

1

)π

(r

1

)(v) = π

(r

1

)π

(1 b/r

1

)(v) = ψ(ab/r) π

(r

1

)(v)

for all b ∈ B and hence π(r

1

)v ∈ Va/r, and it is easy to see that π

(r

1

)is bijective

from Va to Va/r. Since, for all r ∈ F×, π(r

1

)is a vector space isomorphism which

preserves the isotypic subspaces (as sets), there is an action of F× on the set of isotypic

subspaces such that r · Va = Va/r.

This action is almost transitive—almost, because it fixes V0. There is a special class

of representations for which this problem goes away:

Definition 3.24. A representation (π, V ) of GL(2, F ) is cuspidal if V0 = 0. Equiv-

alently, (π, V ) is cuspidal if and only if there is no v ∈ V with π(

1 b1

)(v) = v for all

b ∈ F .

In particular, if (π, V ) is cuspidal, then it is the direct sum of q − 1 isomorphic

subspaces, so q − 1 | dimV .

Proposition 3.25. If E is a field and χ is a character of E× such that χ(E×1 ) is not

trivial, then the representation θ(χ) constructed in section 3.5 is cuspidal.


Proof. We can find c ∈ E×1 such that χ(c) 6= 1. Let Φ ∈ W (χ) and observe that

Φ(0) = χ(c)−1Φ(0), so Φ(0) = 0. Suppose there exists e ∈ E× such that Φ(e) 6= 0.

Then N(e) 6= 0, so ψ(bN(e)

)6= 1 for some b ∈ F× since ψ is nontrivial. Thus(

θ(χ)(

1 b1

)Φ)(e) = ψ

(bN(e)

)Φ(e) 6= Φ(e), so Φ /∈ W (χ)0. In other words, W (χ)0 =

0.

Corollary 3.26. Under the same assumptions, the representation θ(χ) is irreducible.

Proof. As remarked in Proposition 3.25, Φ(0) = 0 for all Φ ∈ W (χ), and if e ∈ E×

then Φ(e) determines the values of Φ on the coset eE×1 , by definition of W (χ). Thus

dimW (χ) = [E× : E×1 ] = |F×| = q− 1, since kerN = E×1 and imN = F× (considering

N as a homomorphism E× → F×). It is clear from the second characterisation in

Definition 3.24 that a subrepresentation of a cuspidal representation is also cuspidal,

so any subrepresentation of θ(χ) must have dimension 0 or q − 1, which means that

θ(χ) is irreducible.

3.7 Inequivalence of cuspidal representations

We saw in Proposition 3.17 that the principal series representations B(χ1, χ2) are pair-

wise inequivalent, up to swapping χ1 and χ2. We would like to prove a similar result

for the cuspidal representations θ(χ) studied in section 3.6. Unfortunately, Mackey

theory is of no use here, because the representations θ(χ) are not constructed as in-

duced representations. We take instead a more elementary approach, considering the

conditions that a nonzero intertwining map must satisfy.

Proposition 3.27. Let χ1 and χ2 be characters of E× such that neither χ1(E×1 ) nor

χ2(E×1 ) is trivial. Construct the representations(θ(χk),W (χk)

)for k = 1, 2 as in

section 3.5. Suppose that T : W (χ1) → W (χ2) is a nonzero intertwiner from θ(χ1) to

θ(χ2). Then either χ2 = χ1 or χ2 = χ′1, where χ′1(u) = χ1(u) for all u ∈ E×.

Proof. In order to make calculations with T , it is easiest to use explicit bases of W (χ1)

and W (χ2). Recall our earlier remarks that if Φ ∈ W (χk) (k = 1 or 2), then Φ(0) = 0;

furthermore, the values of Φ on the coset cE×1 are completely determined by Φ(c), for

c ∈ E×, and are independent of the values of Φ on the other cosets of E×1 . Thus a set

of q−1 functions Φ, each supported on a different coset of E×1 , form a basis for W (χk).

To be precise, fix c1, c2, . . . , cq−1 ∈ E×, all in different cosets of E×1 (in other words,

all with different norms). Then, for k = 1, 2, we define a basis {Φk1, . . . ,Φ

kq−1} of W (χk)

by

Φki (cj) = δij

for 1 ≤ i, j ≤ n, where δij is the Kronecker delta symbol. (Note that Φ1i 6= Φ2

i , in

general, because of the different conditions W (χ1) and W (χ2) place on their members.)

This definition has the advantage that the coefficient of Φki in the function Φ ∈ W (χk)

is easily calculated to be Φ(ci).

3.7. Inequivalence of cuspidal representations 35

For example, we can calculate(θ(χk)

(1 b

1

)Φki

)(cj) = ψ

(bN(cj)

)Φki (cj) = ψ

(bN(cj)

)δij,

and so θ(χk)(

1 b1

)Φki = ψ

(bN(ci)

)Φki for 1 ≤ i ≤ q − 1 and k = 1, 2. Similarly,(

θ(χk)

(a

1/a

)Φki

)(cj) = Φk

i (acj) =

{χk(ci/acj) if N(acj) = N(ci),

0 otherwise,

which means that θ(χk)( a

1/a

)Φki = χk(ci/acj)Φ

kj , where j is chosen such that N(cj) =

N(ci/a). Another easy calculation is that if a ∈ F× and N(c) = a, then(θ(χk)

(a

1

)Φki

)(cj) = χk(c)Φ

ki (ccj) =

{χk(ci/cj) if N(ccj) = N(ci),

0 otherwise,

so θ(χk)(a

1

)Φki = χk(ci/cj)Φ

kj , where j is chosen such that N(cj) = N(ci)/a. Finally,

we consider w:(θ(χk)

(1

−1

)Φki

)(cj) = δq−1

∑u∈V

ψ(tr(ucj)

)Φki (u)

= δq−1∑l∈E×1

ψ(tr(lcicj)

)Φki (lci)

= δq−1∑l∈E×1

ψ(tr(lcicj)

)χk(l)

−1,

so

θ(χk)

(1

−1

)Φki = δq−1

q−1∑j=1

(∑l∈E×1

ψ(tr(lcicj)

)χk(l)

−1

)Φkj .

We want to find the matrix of T with respect to these bases. Let tij ∈ C be such

that

TΦ1i =

q−1∑j=1

tijΦ2j .

for 1 ≤ i, j ≤ q − 1. Since T is an intertwiner, we have

T ◦ θ(χ1)(g) = θ(χ2)(g) ◦ T (3.11)

for all g ∈ GL(2, F ). We first apply (3.11) to g =(

1 b1

)and evaluate at Φ1

i , obtaining

ψ(bN(ci)

) q−1∑j=1

tijΦ2j =

q−1∑j=1

tijψ(bN(cj)

)Φ2j

for all b ∈ F and all i. Since the Φ2j are linearly independent, this implies that

tijψ(bN(ci)

)= tijψ

(bN(cj)

)


for all b ∈ F and all i, j. If i 6= j then N(ci) 6= N(cj), so by Lemma 3.23 ψ(bN(ci)

)6=

ψ(bN(cj)

)for some b ∈ F , and so tij = 0. On the other hand, no condition on tii is

implied.

Now we apply (3.11) to g =(a

1

)for a ∈ F×, obtaining

tjjχ1

(cicj

)= tiiχ2

(cicj

)(3.12)

whenever N(cj) = N(ci)/a. Since a is also arbitrary, this means (3.12) holds for all i.

Since T 6= 0, this means that tii = χ1(ci)/χ2(ci) for all i (up to scaling T by a nonzero

constant, which is irrelevant to us). But (3.11) with g =( a

1/a

)yields

tjjχ1

(ciacj

)= tiiχ2

(ciacj

),

if a ∈ F×, where j is chosen such that N(cj) = N(ci/a). Combined with our knowledge

of tii and tjj, this implies that χ1(a) = χ2(a) for all a ∈ F×, which is at least a step in

the right direction.

However, there are many characters which extend χ1|F× (by the argument in Propo-

sition 3.7). To obtain the conclusion we want, we need to apply (3.11) to g =(

1−1

).

Taking the coefficient of Φkj on both sides, this yields:

χ1(cj)

χ2(cj)

∑l∈E×1

ψ(tr(lcicj)

)χ1(l)−1 =

χ1(ci)

χ2(ci)

∑l∈E×1

ψ(tr(lcicj)

)χ2(l)−1

for all i, j. Grouping the characters χk and observing that l−1 = l since N(l) = 1, we

obtain ∑l∈E×1

ψ(tr(lcicj)

)(χ1

(cjl

ci

)− χ2

(cjl

ci

))= 0

for all i, j. Multiplying by χ1(cici) (which equals χ2(cici) since cici ∈ F×), we see that∑l∈E×1

ψ(tr(lcicj)

)(χ1(cjcil)− χ2(cjcil)

)= 0,

and summing over l instead of l, this becomes∑l∈E×1

ψ(tr(lcjci)

)(χ1(lcjci)− χ2(lcjci)

)= 0

for all i, j. Since N(cjci) takes each value in F× for some i, j, our equation simply says

that ∑m∈E×N(m)=f

ψ(tr(m)

)(χ1(m)− χ2(m)

)= 0

3.7. Inequivalence of cuspidal representations 37

for all f ∈ F×. The substitution m = am′ for a ∈ F× yields∑m′∈E×N(m′)=f

ψ(a tr(m′)

)(χ1(m′)− χ2(m′)

)= 0 (3.13)

for all a, f ∈ F×, after dividing by χ1(a) (which equals χ2(a)).

It’s not immediately obvious what the next step should be, since we have no real

information about χ1, χ2 and ψ. In previous sections, when evaluating sums involving

characters, we have either changed the summation index or used orthogonality rela-

tions. It is not clear that changing the summation index any further would reveal

anything new, but using orthogonality of characters appears promising. If we could

say the additive characters a 7→ ψ(a tr(m′)

)were orthogonal for different m′ (such that

N(m′) = f) then we might be able to conclude that χ1(m′) = χ2(m′) for all relevant m′.

However, the characters a 7→ ψ(a tr(m′)

)are not even all distinct, because if m′ /∈ F×

then m′ and m′ are distinct elements with the same norm and trace. (Of course, that

method could never have worked, because the conclusion that χ1(m′) = χ2(m′) is too

strong.)

A better idea along the same lines is to consider all the possible values of tr(m′),

and make that into the summation index. Because the multiplicative group of every

finite field is cyclic, we can choose a generator γ of E×, and it will suffice to prove the

claim for χ2(γ), since that determines the whole of χ2. Let

S ={p ∈ F

∣∣ p = tr(lγ) for some l ∈ E×1}.

Luckily, this choice simplifies the summation considerably, because of the following

fact: if p ∈ S then there are exactly two l ∈ E×1 such that tr(lγ) = p. To see this,

observe that tr(lγ) = p if and only if

lγ + l−1γ = p (remembering that l = l−1)

⇐⇒ γl2 − pl + γ = 0.

The discriminant of this quadratic equation in l is p2 − 4γγ, which is nonzero for the

following reason: γγ = γq+1 is a generator of F× and hence not a square in F×, because

|F×| = q−1 is even. Therefore there are either zero or two solutions for l, and so there

must be two solutions if p ∈ S; the other solution is γ/γl, which is also in E×1 .

For each p ∈ S choose any l(p) such that tr(γl(p)

)= p. The two terms in the

summation in (3.13) whose ψ factor is ψ(ap) are those with l = l(p) and l = γ/γl(p) =

γl(p)/γ. Thus we can rewrite (3.13) as∑p∈S

ψ(ap)(χ1

(l(p)γ

)− χ2

(l(p)γ

)+ χ1

(l(p)γ

)− χ2

(l(p)γ

))= 0 (3.14)

for all a ∈ F×. But χ1 and χ2 are nontrivial on E×1 , so∑p∈S

(χk(l(p)γ

)+ χk

(l(p)γ

))=∑l∈E×1

χk(lγ)

= 0


for k = 1, 2, by the same character trick we used before. Thus (3.14) is also true in

the case a = 0, so it is true for all a ∈ F . Now we know the characters a 7→ χ(ap)

for p ∈ S are distinct, hence orthogonal, hence linearly independent. It follows that

the coefficients of ψ(ap) in (3.14) are all zero. In particular, when p = tr(γ), we may

choose l(p) = 1 and hence

χ1(γ) + χ1(γ)− χ2(γ)− χ2(γ) = 0.

But we also have χ1(γ)χ1(γ) = χ2(γ)χ2(γ), since χ1 and χ2 agree on F×. So the

pairs {χ1(γ), χ1(γ)} and {χ2(γ), χ2(γ)} are both the solution set of the same quadratic

equation, so either χ2(γ) = χ1(γ) or χ2(γ) = χ1(γ). Since γ generates E×, we now

know the relationship: in the first case χ2 = χ1, and in the second case χ2 = χ′1.

The above proposition shows that the representations θ(χ) (where χ is a character

of E× such that χ(E×1 ) is nontrivial) are all inequivalent, except for the “conjugate

pairs” of characters χ and χ′ described above.

For completeness, we should check that θ(χ) is equivalent to θ(χ′), where χ′(x) =

χ(x) for all x ∈ E×. It is easily checked that the map T : W (χ)→ W (χ′) as (TΦ)(u) =

Φ(u) for all u ∈ E is an intertwiner from θ(χ) to θ(χ′). However, in the following

paragraph this check is rendered unnecessary, because if θ(χ) were not equivalent to

θ(χ′), there would be too many irreducible representations of GL(2, F ).

The natural question now is whether there are any more irreducible representations

of GL(2, F ). Recalling a lemma from finite group representation theory, it will be

enough to check that the sum of the squares of the dimensions of the inequivalent

irreducible representations we know about is equal to the order of the group. Here are

the irreducible representations we know about, in summary form:

Type Number Dimension∑

Dimension2

Character q − 1 1 q − 1

Special q − 1 q q3 − q2

Principal series 12(q − 1)(q − 2) q + 1 1

2(q4 − q3 − 3q2 + q + 2)

Cuspidal 12q(q − 1) q − 1 1

2(q4 − 3q3 + 3q2 − q)

q4 − q3 − q2 + q

To find the order of GL(2, F ), we need only observe that, in order to obtain an

invertible matrix, the first column can be any of q2 − 1 nonzero vectors, and once the

first column is chosen, the second column can be any of q2 − q vectors which are not

collinear with the first column. So |GL(2, F )| = (q2 − 1)(q2 − q) = q4 − q3 − q2 + q,

which shows that we have found all the irreducible representations of GL(2, F ).

Chapter 4

Representations over local fields

In this chapter, we modify our previous techniques in order to find representations of

GL(2, F ), where F is a non-Archimedean local field with odd residue characteristic. We

first outline the definition and properties of local fields, introduce the appropriate no-

tion of admissible representations, and give some background on the Fourier transform

over local fields. We then quote a theorem of Segal, Shale and Weil, which provides

a family of intertwiners corresponding to elements of SL(2, F ) in the same way as

the Finite Stone–von Neumann Theorem did in the finite field case. The proof of the

theorem is beyond the scope of this essay. We construct a projective representation of

SL(2, F ), and lift it to a true representation of SL(2, F ) called the Weil representation.

We obtain a model of this representation which operates very similarly to the finite

field case.

Finally, by specialising the parameters of our representation, we obtain a family of

representations of GL(2, F )+, where GL(2, F )+ is a certain subgroup of GL(2, F ) of

index two. We then use the induced representation to find representations of GL(2, F ).

The reason we cannot obtain a representation of GL(2, F ) directly is that the norm

map from a quadratic extension of F is no longer surjective in the local field case. The

representations we obtain are known as the dihedral supercuspidal representations.

4.1 Local fields

There are multiple definitions of a local field; we will choose one of the simpler defini-

tions, and piece together the most important properties.

Definition 4.1. A field F , together with a topology T on F , is called a topological

field if the mapsF × F → F : (x, y) 7→ x+ y,

F × F → F : (x, y) 7→ xy,

and F \ {0} → F : x 7→ x−1

are continuous (with respect to the product topology on F ×F ). The topological fields

F and G are topologically isomorphic if there exists a field isomorphism F → G

39

40 Chapter 4. Representations over local fields

which is also a homeomorphism.

If F is Hausdorff, locally compact and not discrete, then we call F a local field.

While the topological definition is very general, it turns out that local fields can be

analysed in terms of the more concrete concept of an absolute value.

Definition 4.2. An absolute value on a field F is a map ϕ : F → R which satisfies

the following conditions:

(a) ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∈ F ;

(b) ϕ(a) ≥ 0 for all a ∈ F , with equality if and only if a = 0; and

(c) ϕ(a+ b) ≤ ϕ(a) + ϕ(b) for all a, b ∈ F .

Clearly an absolute value ϕ on F induces a metric d on F given by d(a, b) = ϕ(a−b),and hence a topology on F .

Now we can state the classification theorem for local fields.

Theorem 4.3. Every local field is topologically isomorphic to one of the following:

• R or C, with the topology induced by the standard absolute value;

• the p-adic numbers Qp for some prime p, with the topology induced by the

standard p-adic absolute value, or a finite extension of Qp, with the product

topology as a vector space over Qp; or

• the field Fq((t)) of formal Laurent series (that is, formal power series in t and t−1

with finitely many terms of negative degree) over some finite field Fq, with the

topology induced by the absolute value ϕ(∑

n∈Z antn)

= q−k, where k is the

smallest integer such that ak 6= 0.

We prove this theorem through a number of lemmas. We first need a slight weak-

ening of the conditions on an absolute value.

Lemma 4.4. Suppose that ψ : F → R satisfies (a) and (b) above, and that in addition

(d) there exists some C ∈ R such that ψ(1 + a) ≤ C whenever ψ(a) ≤ 1.

Then there is some λ > 0 such that ψλ is an absolute value on F .

Proof. See [2], Lemmas 1.1 and 1.2 on p. 13.

Lemma 4.5. Let F be a local field with topology T . There exists a continuous map

ψ : F → R which satisfies (a), (b) and (d) above. Moreover, S = {a ∈ F | ψ(a) ≤ 1}is compact. In addition, T = Tϕ, where ϕ is any absolute value obtained from ψ by

Lemma 4.4 and Tϕ is the topology on F induced by ϕ. In other words, the topology

on F is induced by an absolute value.

4.1. Local fields 41

Proof. See [11], Lemmas 1, 2 and 3 on pp. 369–70. The essential step is to take a

Haar measure µ on the Borel σ-algebra of the additive group F . Then for all a ∈ F×the measure µa, defined by µa(X) = µ(aX) for Borel sets X, is also an additive Haar

measure on F . Since the Haar measure is unique up to scalar multiple, there is some

ψ(a) ∈ R such that µ(aX) = ψ(a)µ(X) for all Borel sets X and all a ∈ F×. Setting

ψ(0) = 0, it can then be checked that ψ is continuous and satisfies (a), (b) and (d).

For the last claim, what is proved in [11] is that T = Tψ, where Tψ denotes the

topology given by the base of neighbourhoods B(x, r) = {y ∈ F | ψ(x− y) < r} for all

x ∈ F and r ∈ R+. Since ϕ = ψλ for some λ > 0, we have Tψ = Tϕ.

The next lemma allows us to associate to each local field a unique absolute value,

up to equivalence, which in this context means raising the absolute value to a positive

power.

Lemma 4.6. If ϕ1 and ϕ2 are two absolute values on a local field F which both induce

the same topology, then ϕ1 = ϕλ2 for some λ > 0.

Proof. See [2], Lemma 3.2 on p. 20.

The essential feature determining the type of a local field is whether or not its

absolute value ϕ satisfies the ultrametric inequality, which states that ϕ(u+v) ≤max{ϕ(u), ϕ(v)} for all u, v ∈ F . It is easy to see that this is a property of the local field

itself, since the ultrametric inequality is preserved among equivalent absolute values.

In reference to the Archimedean property of R (namely, that for every r ∈ R there is

an integer n with |n| > |r|), local fields where the ultrametric inequality fails are called

Archimedean. All other local fields are called non-Archimedean.

Lemma 4.7. Let F be a local field with topology T . Then F is complete with respect

to any absolute value which induces the topology T .

Proof. Let ϕ be an absolute value on F which induces the topology T . Let (xn)n∈N

be a Cauchy sequence in F with respect to ϕ. Since this sequence is Cauchy, it is

bounded, so there exists some M > 0 such that ϕ(xn) < M for all n ∈ N. Picking

any b ∈ F such that ϕ(b) > M , we see that xn ∈ {a ∈ F | ϕ(a) ≤ ϕ(b)} = bS for all

n ∈ N, and bS is compact since S is compact (by Lemma 4.5) and multiplication is

continuous. Since bS is compact, it is complete with respect to ϕ, so xn converges to

some x ∈ bS.

Theorem 4.8 (Ostrowski). Let F be an Archimedean local field. Then F is topolog-

ically isomorphic either to R or to C.

Proof. See [2], Theorem 1.1 on p. 33. We first need to apply Lemma 4.7 above to see

that F is complete with respect to the absolute value we use on it.

From this point on, we will consider only non-Archimedean local fields.


Definition 4.9. Let F be a non-Archimedean local field with absolute value ϕ. Then

the definition of an absolute value and the ultrametric inequality imply that

o :={x ∈ F

∣∣ ϕ(x) ≤ 1}

is a subring of F , called the ring of integers of F . Since ϕ(a)ϕ(1/a) = 1 for all

a ∈ F×, the elements of o of absolute value 1 are invertible in o. It follows that

p :={x ∈ F

∣∣ ϕ(x) < 1}

is a maximal ideal of F , and o/p is a field, called the residue class field of F . The

characteristic of o/p is known as the residue characteristic of F .

Remark 4.10. Since the elements of absolute value 1 are invertible in o, any proper

ideal of o is contained in p. Thus p is the unique maximal ideal of o.

Proposition 4.11. Let F , ϕ, o and p be as above. Then o and p are open and compact,

and o/p is finite. Moreover, there exists $ ∈ p such that p = $o, ϕ($) ∈ (0, 1) and

imϕ = {0} ∪ {ϕ($)n | n ∈ Z}.

Proof. By Lemma 4.5, o is compact. Since ϕ is continuous and p is the preimage of

the open set (−∞, 1) under ϕ, p is open. We see that

F \ p =⋃

x∈F\p

(x+ p) and o =⊔

x+p∈ o/p

(x+ p)

which shows that p is closed, hence compact, and o is open. Moreover, since the union

is disjoint, all the sets x+ p are needed to cover o. Since o is compact, this means that

o/p is finite. Let q = |o/p|.Observe that ϕ(p) is the continuous image of a compact set and is therefore compact,

but also that ϕ(p) ⊆ [0, 1). Let α = supp∈p ϕ(p). Since p is compact, there exists some

$ ∈ p such that ϕ($) = α, so α < 1. But α 6= 0 (otherwise ϕ(p) = 0 for all

p ∈ p, which would mean that p = {0} and thus that F were discrete, contrary to

assumption). So α ∈ (0, 1).

For any x ∈ F×, let k = min{n ∈ Z | $nx ∈ o}. Then ϕ($kx) ∈ (α, 1] by the

minimality of k, so ϕ($kx) = 1 since ϕ takes no values in the range (α, 1). Thus

ϕ(x) = ϕ($−k) = α−k. Since also ϕ(0) = 0, we have imϕ = {0} ∪ {αn | n ∈ Z}.

Remark 4.12. Every non-Archimedean local field is totally disconnected. To see this,

using the notations above, suppose x, y ∈ F and x 6= y. If k is such that ϕ(x−y) = αk,

then x+$k+1o is an open and closed set which contains x but not y.

Definition 4.13. Let F , o, p and $ be as above, and let q = |o/p|. The standard

absolute value on F is the absolute value ϕ satisfying ϕ($) = 1/q.

4.1. Local fields 43

Henceforth we will always use the standard absolute value on each non-Archimedean

local field, and we will denote it by x 7→ |x|. One reason why this is a natural choice

is that the additive Haar measure µ on F then satisfies

µ(aS) = |a|µ(S) (4.1)

for any Borel subset S of F .

Lemma 4.14. Let F , o, p and $ be as above, and let S be a complete set of coset

representatives for p in o, including 0. Then every x ∈ F can be written uniquely as a

convergent series of the form

x =∑n∈Z

sn$n

for some coefficients sn ∈ S such that sn = 0 for all but finitely many negative n.

Proof. By multiplying by a suitable power of $, it will suffice to prove that every x ∈ o

can be written uniquely in the above form. We prove by induction on k that there exists

a unique choice of s0, s1, . . . , sk ∈ S such that x−∑k

n=0 sn$n ∈ $k+1o for all k ≥ −1.

For the k = −1 case, we already have x ∈ o. Now, supposing the result true for k, by

definition of S there exists a unique sk+1 ∈ S such that x−∑k+1

n=0 sn$n ∈ $k+1o.

Clearly the residue characteristic p of F always divides the characteristic of F , so

charF = 0 or charF = p. The proof of Theorem 4.3 is completed by the following

proposition.

Proposition 4.15. Let F be a non-Archimedean local field with residue characteris-

tic p, and let o, p and q be as above. If charF = p, then F is isomorphic to Fq((t)). If

charF = 0, then F is isomorphic to a finite extension of Qp.

Proof. If charF = p, then o is isomorphic to the ring Fq[[t]] of formal power series over

the finite field of order q—for the proof, see [8], Theorem 2 on p. 33. F is clearly then

isomorphic to the field of fractions of Fq[[t]], namely Fq((t)).If charF = 0, then there is a natural injective ring homomorphism i : Z → o

determined by 1 7→ 1. We note that i(p) + p = 0 in o/p, which means that |i(p)| < 1.

In addition, |i(a)− i(b)| = |i(p)|k if |a− b|p = p−k, which means that Cauchy sequences

in Z with respect to the p-adic metric are mapped to Cauchy sequences in o. Since Zp

is the completion of Z under the p-adic metric, there exists a unique continuous map

j : Zp → o extending i, and j is clearly an injective ring homomorphism. Since F is a

field, there exists a unique embedding : Qp → F extending j.

We know q = pk for some k ∈ N. Pick any k-tuple (b1 = 1, b2, · · · , bk) ⊆ ok

such that (b1 + p, · · · , bk + p) forms a basis for o/p over its prime subfield. Then

S ={j(a1)b1 + · · · + j(ak)bk

∣∣ ai ∈ {0, . . . , p − 1}}

forms a complete set of coset

representatives for p in o. Since |i(p)| < 1, we can find l ≥ 1 and s ∈ S such that

i(p)−$ls ∈ $l+1o. By Lemma 4.14, every x ∈ F can be written uniquely in the form

x =∑

n∈Z sn$n for some coefficients sn ∈ S. This can be uniquely decomposed in the

form x =∑l−1

n=0

∑ki=1 (ain)bi$

n for some ain ∈ Qp. Thus [F : (Qp)] = lk, so F is

isomorphic to a finite extension of Qp.


The representation theory of Lie groups like GL(2,R) and GL(2,C) uses techniques

from Lie theory (see [1], Chapter 2, and [9], Chapter V for SL(2,R)). By contrast,

the representation theory of GL(2, F ), where F is a non-Archimedean local field, has

strong parallels to the finite field theory, and that is what we will study in this chapter.

4.2 Smooth and admissible representations

It is not generally practical to study arbitrary representations of infinite topological

groups, because the topology on the group is of no use in that case. Very often,

attention is restricted to unitary representations, which are homomorphisms from a

group into the group of unitary operators on a Hilbert space which satisfy a certain

strong continuity property (see [9], Chapter I). We will, however, find it more convenient

to operate within the framework of smooth and admissible representations, which have

the advantage of requiring no topology on the representation space.

We first state a lemma, useful in its own right, which illustrates how we can define

a useful substitute for continuity, called smoothness, without using any topology on

the target vector space.

Lemma 4.16. Let G be a totally disconnected locally compact group and let ψ be a

character of G. Then ψ is locally constant; that is, every g ∈ G has a neighbourhood

on which ψ is constant.

Proof. It will suffice to prove that ψ is constant on an open neighbourhood of the

identity, because ψ is then constant on any translation of that neighbourhood. By

the general theory of totally disconnected groups, any open neighbourhood of the

identity in G contains a compact open subgroup. Consider the open neighbourhood

ψ−1{x ∈ C× | Rex > 0} of the identity. It contains a compact open subgroup U . Then

ψ(U) is a subgroup of T all of whose elements have positive real parts; thus it is the

trivial subgroup, and so ψ is constant on U as claimed.

It is too much to ask that a representation itself be locally constant, but we instead

ask that the group’s action on each individual vector in the representation space be

locally constant.

Definition 4.17. Let G be a totally disconnected locally compact group, V a complex

vector space and π : G→ GL(V ) a homomorphism. Then π is a smooth represen-

tation if for any v ∈ V the stabiliser Gv = {g ∈ G | π(g)v = v} is open. If in addition

the subspace {v ∈ V | π(h)v = v for all h ∈ H} is finite dimensional for every open

subgroup H ≤ G, then π is an admissible representation.

Occasionally we will need to use a unitary representation, which means a

representation ρ : G → U(H), for some Hilbert space H, where U(H) denotes the

group of unitary operators on H.

We will primarily be considering admissible representations of GL(2, F ).

4.3. Fourier analysis on local fields 45

4.3 Fourier analysis on local fields

In order to find representations of GL(2, F ), we will need to understand the properties

of Fourier transforms over local fields. The theory is developed similarly to Fourier

theory on R, except that some of the arguments are a little simpler in the local field

case.

Let F be a non-Archimedean local field, o the ring of integers of F , p the unique

maximal ideal of o, $ any generator of p, and | · | the standard absolute value on F .

Let ψ be a fixed nontrivial additive character of F . By Lemma 4.16, there exists some

n ∈ Z such that ψ is trivial on $no. Let c ∈ Z be the least such n; $co is then called

the conductor of ψ.

Let µ be an additive Haar measure on F , normalised so that µ(o) = 1.

Lemma 4.18. For any n ∈ Z and t ∈ F ,∫$no

ψ(tx) dµ(x) =

{q−n if |t| ≤ qn−c;

0 otherwise.

Proof. If |t| ≤ qn−c then ψ(tx) = 1 for all x ∈ $no. Otherwise, take any y ∈ $c−1o

such that ψ(y) 6= 1. Then observe that y/t ∈ $no, so by the translation-invariance

of µ, ∫$no

ψ(tx) dµ(x) =

∫$no

ψ(t(x+ y/t)

)dµ(x) = ψ(y)

∫$no

ψ(tx) dµ(x).

Definition 4.19. Let V be a finite-dimensional vector space over F . We define the

Schwartz space S(V ) of V to be the complex vector space of locally constant func-

tions f : V → C of compact support.

Definition 4.20. Let f ∈ S(F ). The Fourier transform f of f is defined by

f(t) =

∫F

f(x)ψ(tx) dµ(x)

for all t ∈ F .

If f ∈ S(F ), there exist m,n ∈ Z such that supp f ⊆ $mo and f is constant

on cosets of $no in F . Thus f is bounded. It is easily seen from Lemma 4.18 that

f(t) = ‖f‖1 for sufficiently small t and f(t) = 0 for sufficiently large t. Thus f ∈ S(F ).

Lemma 4.21. Let f, g ∈ S(F ). Then∫F

f(x)g(x) dµ(x) =

∫F

f(y)g(y) dµ(y).

Proof. For all x, y ∈ F , |f(y)g(x)ψ(xy)| ≤ ‖f‖∞‖g‖∞, so∫F

(∫F

|f(y)g(x)ψ(xy)|dµ(y)

)dµ(x) ≤ µ(supp f)µ(supp g)‖f‖∞‖g‖∞ <∞.


Thus by Fubini’s theorem∫F

(∫F

f(y)ψ(xy) dµ(y)

)g(x) dµ(x) =

∫F

f(y)

(∫F

g(x)ψ(xy) dµ(x)

)dµ(x).

Lemma 4.22. For all n ∈ Z we have 1$no = q−n1$c−no.

Proof. This is a direct consequence of Lemma 4.18.

For any f : F → C and y ∈ F the translate fy is defined by fy(x) = f(x− y) for

all x ∈ F . From the definition of the Fourier transform, we obtain fy(t) = ψ(ty)f(t).

Proposition 4.23 (Fourier inversion formula). Let f ∈ S(F ). Then

ˆf(x) = q−cf(−x)

for all x ∈ F .

Proof. The following integrals estimateˆf(x): if n ∈ Z,∫

F

f(y)ψ(xy)1$−no(y) dµ(y) =

∫F

fx(y)1$−no(y) dµ(y) (4.2)

=

∫F

fx(y)1$−no(y) dµ(y) by Lemma 4.21

= qn∫$c+no

f(y − x) dµ(y) by Lemma 4.22

= q−cf(−x) for sufficiently large n.

But the integrand on the left of (4.2) is dominated for all n ∈ Z by∣∣f ∣∣, which is

integrable. By the Dominated Convergence Theorem, as n→∞ we have∫F

f(y)ψ(xy)1$−no(y) dµ(y)→∫F

f(y)ψ(xy) dµ(y) =ˆf(x).

Proposition 4.24. If f ∈ S(F ), then ‖f‖22 = q−c‖f‖2

2.

Proof. Denote by f the function x 7→ f(x). It follows from the definition of the Fourier

transform that f(x) = f(−x) for all x ∈ F . Thus

‖f‖22 =

∫F

f(x)f(x) dµ(x) =

∫F

f(x)f(−x) dµ(x)

=

∫F

f(x)f(−x) dµ(x) = q−c

∫F

f(x)f(x) = q−c‖f‖22.

We need to consider Fourier transforms on larger spaces than the Schwartz space.

We will focus on L2 spaces.


Definition 4.25. Let V be a finite-dimensional vector space over F and let ν be a Haar

measure on V . For any measurable functions f, g : V → C we define the convolution

f ∗ g of f and g with respect to the measure ν by

(f ∗ g)(x) =

∫V

f(x− y)g(y) dν(y)

for all x such that the integral is defined.

Proposition 4.26. The Schwartz space S(F ) is dense in L2(F ). For any finite-

dimensional vector space V over F , the Schwartz space S(V ) is dense in L2(V ).

Proof. Let f ∈ L2(F ). Then f ·1K ∈ L1 for all compact K ⊆ F by the Cauchy-Schwarz

inequality. For each n ∈ N let fn = 1$−no · (qn1$no ∗ f). We note the convolution

(qn1$no ∗ f)(x) is the integral of f over a compact subset of F , so fn(x) is defined for

all x. Since 1$no(x + t) = 1$no(x) for all x ∈ F and t ∈ $no, we have fn ∈ S(F ).

We claim fn → f in L2(F ) as n → ∞. Indeed, fix ε > 0. By the continuity of

translation ([4], Proposition 2.41), there exists n0 ∈ N such that ‖f − fy‖22 < ε/2

whenever y ∈ $n0o. By the Dominated Convergence Theorem, there exists m0 ∈ Nsuch that

∫F\$−no

|f |2 dµ < ε/2 for all n > m0. Thus for all n > max{n0,m0} we have

∫F

|f − fn|2 dµ =

∫F\$−no

|f |2 dµ+

∫$−no

∣∣∣∣f(x)− qn∫$no

f(x− y) dµ(y)

∣∣∣∣2dµ(x)

<ε

2+ q2n

∫$−no

∣∣∣∣∫$no

(f(x)− f(x− y)

)dµ(y)

∣∣∣∣2dµ(x)

1=ε

2+ q2n

∫$−no

q−n∫$no

∣∣f(x)− f(x− y)∣∣2 dµ(y) dµ(x)

2=ε

2+ qn

∫$no

∫$−no

∣∣f(x)− fy(x)∣∣2 dµ(x) dµ(y)

≤ ε

2+ qn

∫$no

ε

2dµ(y) = ε,

where we used the Cauchy-Schwarz inequality at step 1 and Fubini’s theorem at step 2.

The statement for S(V ) is proved by a very similar argument in multiple dimensions,

replacing $no by a small compact open subgroup of V .

4.4 A projective representation of SL(2, F )

For the remainder of this chapter, we will take F to be a non-Archimedean local field

with odd residue characteristic. Let o be the ring of integers of F , p the unique

maximal ideal of o, $ any generator of p, and | · | the standard absolute value on F .

As in the finite field case, let V be a vector space over F of finite dimension d, and B a

nondegenerate symmetric bilinear form on V . Let H be the corresponding Heisenberg

group, as defined in Definition 2.11. Let ψ be a fixed nontrivial additive character of F .


Our aim is to obtain representations of GL(2, F ) by roughly the same steps as we

used in the finite field case: first obtain a projective representation of SL(2, F ), lift to

a true representation of SL(2, F ), then extend that representation to a representation

of GL(2, F ). There are a few more obstacles along this path in the local field case, but

the broad outline and conclusions will be very similar.

The first obstacle we encounter is that the finite Stone–von Neumann Theorem

of Section 3.1 only applies to finite groups. We will state a theorem of Segal, Shale

and Weil, which provides a similar result for a special class of locally compact abelian

groups. (The proof of the theorem is beyond the scope of this essay.) This special class

does not contain the Heisenberg group H, but it does contain a closely related group,

A(V ), which we will define next.

Definition 4.27. Let A(V ) be the set V × V × T, equipped with the multiplication

(u, v, t)(u′, v′, t′) =(u+ u′, v + v′, tt′ψ(−2B(v, u′))

).

We can check this is a group: associativity follows from the fact that

ψ(−2B(v, u′)

)ψ(−2B(v + v′, u′′)

)= ψ

(−2B(v, u′ + u′′)

)ψ(−2B(v′, u′′)

)for all u, u′, u′′, v, v′, v′′ ∈ V ; the identity element is (0, 0, 1); and the inverse of (u, v, t)

is(−u,−v, t−1ψ(−2B(v, u))

).

Remark 4.28. The group A(G) can actually be defined for any locally compact abelian

group G, by taking the underlying set to be G∗ × G × T, where G∗ is the group of

characters of G, and replacing ψ(−2B(v, u′)) in the multiplication rule with u′(v). We

will not need this level of generality.

There is a unitary representation ρ of A(V ) on L2(V ) given by(ρ(u′, v′, t′)Φ

)(v) = t′ψ

(−2B(v, u′)

)Φ(v + v′)

for all (u′, v′, t′) ∈ A(G) and v ∈ V . That this is in fact a representation reduces to

the same identity as in Definition 4.27. It is easily checked that the representation is

unitary: the operators ρ(u′, v′, t′) are invertible and∫V

∣∣t′ψ(−2B(v, u′))Φ(v + v′)

∣∣2dν(v) =

∫V

∣∣Φ(v + v′)∣∣2dν(v) =

∫V

∣∣Φ(v)∣∣2dν(v)

for all (u′, v′, t′) ∈ A(V ) and Φ ∈ L2(V ), by the translation-invariance of ν.

By an argument similar to that in Remark 2.12, the centre of A(V ) is {(0, 0, t) |t ∈ T}. Let B0 be the group of automorphisms of A(V ) which fix every element of its

centre.

Theorem 4.29 (Segal, Shale and Weil). The representation ρ has no closed invariant

subspaces except for 0 and L2(V ) itself. For each σ ∈ B0(V ) there exists a unitary

operator ω(σ) on L2(V ), unique up to scalar multiple, such that

ρ(σ · h) = ω(σ)ρ(h)ω(σ)−1 (4.3)

for all h ∈ A(V ).


Proof. See [1], Theorem 4.8.3 on p. 531, which follows Weil’s proof in [10].

We would like to transfer this result to H. The first step is to find a homomorphism

τ : H → A(V ). Because of the multiplication laws in H and A(V ) are defined slightly

differently, we need a correction factor in the third coordinate; we define τ by

τ(u, v, x) =(u, v, ψ(x−B(u, v))

)for all (u, v, x) ∈ H. This is indeed a homomorphism, because

τ(u, v, x)τ(u′, v′, x′) =(u, v, ψ(x−B(u, v))

)(u′, v′, ψ(x′ −B(u′, v′))

)=(u+ u′, v + v′, ψ(x+ x′ −B(u, v)−B(u′, v′)− 2B(u′, v)

)= τ(u+ u′, v + v′, x+ x′ +B(u, v′)−B(u′, v)

)= τ((u, v, x)(u′, v′, x′)

)for all (u, v, x), (u′, v′, x′) ∈ H.

Consequently, we have a representation π = ρ◦τ ofH on L2(V ). This representation

is given by the formula(π(u′, v′, x′)Φ

)(v) =

(ρ(u′, v′, ψ(x′ −B(u′, v′))

)Φ)(v)

= ψ(x′ −B(2v + v′, u′)

)Φ(v + v′) (4.4)

for (u′, v′, x′) ∈ H, Φ ∈ L2(V ) and v ∈ V , which is identical to the representation given

by (3.5) that we considered in the finite field case.

Recall the action of SL(2, F ) on H given in of Proposition 2.13. We would like to

use Theorem 4.29 in conjunction with Proposition 2.13 to obtain a projective repre-

sentation of SL(2, F ). In other words, we would like an action of SL(2, F ) on A(V )

by automorphisms which satisfies

g · τ(h) = τ(g · h)

for all g ∈ SL(2, F ) and h ∈ H, which would mean that(a b

c d

)·(u, v, ψ(x−B(u, v))

)= τ(au+ bv, cu+ dv, x)

=(au+ bv, cu+ dv, ψ(x−B(au+ bv, cu+ dv))

)for all

(a bc d

)∈ SL(2, F ) and (u, v, x) ∈ H. This motivates the following action of

SL(2, F ) on A(V ):(a b

c d

)· (u, v, t) :=

(au+ bv, cu+ dv, tψ(B(u, v)−B(au+ bv, cu+ dv))

).

This is indeed an action, because the factor ψ(−B(au + bv, cu + dv)) in the third

coordinate cancels out when we evaluate(a′ b′

c′ d′

)·(a bc d

)· (u, v, x). It is also an action

by group automorphisms, because((a bc d

)· (u, v, t)

)((a bc d

)· (u′, v′, t′)

)=(a bc d

)·(u+ u′, v + v′, tt′ψ(−2B(v, u′))

)


for(a bc d

)∈ SL(2, F ) and (u, v, t), (u′, v′, t′) ∈ A(V ); checking this relation comes down

to the fact that

ψ(B(u, v′)−B(v, u′)−B(au+ bv, cu′ + dv′) +B(au′ + bv′, cu+ dv)

)= ψ

((1− ad+ bc)B(u, v′) + (−1− bc+ ad)B(v, u′)

)= 1.

Note that these automorphisms fix the centre of A(V ), so σg ∈ B0(V ) for all g ∈SL(2, F ), where σg denotes the automorphism of A(V ) determined by g.

We can now use our action to study the representation π of H on L2(V ). Indeed,

from (4.3) there exists a unitary transformation η(g) = ω(σg) of L2(V ), unique up to

scalar multiple, such that

ρ(τ(g · h)

)= ρ(g · τ(h)

)= η(g)ρ

(τ(h)

)η(g)−1

for all g ∈ SL(2, F ) and h ∈ H, or in other words

π(g · h)η(g) = η(g)π(h). (4.5)

By the same argument as in Corollary 3.12, the images of η(g) in PGL(L2(V )

)for

g ∈ SL(2, F ) form a projective representation of SL(2, F ).

The representation π of H on L2(V ) is given by (4.4), which is the same equation

as (3.5) in Section 3.2. Therefore, the reasoning of that section applies unchanged in

this section to calculate the intertwiners η(g) for g ∈ SL(2, F ), except that we must

verify the resulting intertwiners are unitary. Consider any character L• : a 7→ La of F×,

to be specified later. If we define(η

(1 b

1

)Φ

)(v) = ψ

(bB(v, v)

)Φ(v)

and

(η

(a

1/a

)Φ

)(v) = La|a|d/2Φ(av),

for Φ ∈ L2(V ) and v ∈ V , then we claim that η(

1 b1

)and η

( a1/a

)are unitary operators

on L2(V ) satisfying (4.5). They are certainly invertible, since η(

1 b1

)and η

(1 −b

1

)are mutually inverse, and so are η

( a1/a

)and η

(1/a

a

). It is clear that the results

of applying these operators to Φ ∈ L2(V ) are themselves A-measurable and square-

integrable. That η(

1 b1

)is unitary is clear from the fact that

∣∣ψ(bB(v, v))∣∣ = 1 for all

b ∈ F and v ∈ V . That η( a

1/a

)is unitary requires a calculation:∫

V

∣∣La|a|d/2Φ(av)∣∣2dν(v) =

∫V

|Φ(av)|2|a|ddν(v) =

∫V

|Φ(u)|2dν(u),

since ν(aS) = |a|dν(S) for all Borel subsets S of V , where ν is the Haar measure onV ;

this follows from (4.1) and the fact that the product measure of µ in each of the d

dimensions of V gives the Haar measure on V . So η(

1 b1

)and η

( a1/a

)(for a ∈ F×

and b ∈ F ) are the unitary operators determined by (4.5), up to scalars.


Finding η(

1−1

)is not quite as easy as in the finite field case, since we cannot

simply consider every possible operator on the space L2(V ). In the finite field case, we

found that η(

1−1

)was a finite Fourier transform operator. In the local field case, it

turns out that the operator η(

1−1

)is essentially the Fourier transform on L2(V ).

For any Φ ∈ S(V ), we define the Fourier transform Φ : V → V of Φ by

Φ(v) =

∫V

ψ(2B(u, v)

)Φ(u) dν(u) (4.6)

for all v ∈ V . One can easily check that Φ ∈ S(V ). We will occasionally denote Φ by

FΦ. We recall that we have not yet fixed the normalisation of the Haar measure ν, so

we now specify ν to be normalised such that the Fourier inversion formula holds in the

form(FΦ)(v) = Φ(−v) for all Φ ∈ S(V ). The measure ν is then called the self-dual

Haar measure with respect to the pairing (u, v) 7→ ψ(2B(u, v)

), and the Plancherel

theorem holds in the form ‖Φ‖2 = ‖Φ‖2 for all Φ ∈ S(V ). The Plancherel theorem

implies that this map can be extended uniquely to an isometric isomorphism F :

L2(V )→ L2(V ) : Φ 7→ Φ. See [4], Chapter 4, for details of this standard construction.

We set

η

(1

−1

)Φ = KΦ (4.7)

for all Φ ∈ L2(V ), where K ∈ T is a constant yet to be determined. Since F is unitary,

η(

1−1

)is a unitary operator on L2(V ).

To prove η(

1−1

)satisfies (4.5), we note that for all (u′, v′, x′) ∈ H, all Φ ∈ S(V )

and all v ∈ V we have(η(

1−1

)π(u′, v′, x′)Φ

)(v) =

∫V

ψ(2B(u, v)

)ψ(x′ −B(2u+ v′, u′)

)Φ(u+ v′) dν(u)

1=

∫V

ψ(2B(u− v′, v)

)ψ(x′ −B(2u− v′, u′)

)Φ(u) dν(u)

= ψ(x′ −B(2v − u′, v′)

) ∫V

ψ(2B(u, v − u′)

)Φ(u) dν(u)

=(π((

1−1

)· (u′, v′, x′)

)η(

1−1

)Φ)(v),

where we translated the integrand by −v′ at step 1. Since the above equation holds for

all Φ in S(V ), since S(V ) is dense in L2(V ), and since η(

1−1

)and π(h) (for h ∈ H)

are continuous, it follows that π((

1−1

)· h)η(

1−1

)Φ = η

(1

−1

)π(h)Φ for all h ∈ H

and all Φ ∈ L2(V ). We conclude that η(

1−1

)is the operator determined up to scalars

by (4.5).

4.5 A true representation of SL(2, F )

We must again check that the values of the constants in the definition of η can be

chosen such that η becomes a true representation of SL(2, F ). It turns out that this

can only be done if the dimension d of V is even, so we will henceforth assume that d


is even. We will again use the presentation in Proposition 2.15 to check whether η is a

true representation.

Since the representing operators η(g) for g ∈ SL(2, F ) are all continuous and since

S(V ) is a dense subspace of L2(V ), it will suffice to prove the relations hold when

applied to elements of S(V ). This allows us to calculate Fourier transforms directly as

integrals.

By the same calculations as in Section 3.3, the first three relations of (2.2) are

satisfied by the images of t(a) and n(b) under η (for a ∈ F× and b ∈ F ), since we

have already insisted that L• : a 7→ La be a character of F×. Thus we need only check

the fourth and fifth relations, taking into account the new definition of η(

1−1

)as a

Fourier transform.

The fourth relation can be checked in a similar fashion to the finite field case: for

all Φ ∈ S(V ) and v ∈ V ,

(w · t(a) · w · Φ

)(v) = K

∫V

La|a|d/2Kψ(2B(u, v)

)Φ(au) dν(u)

= K2La|a|−d/2∫V

ψ(2B(au, v/a)

)Φ(au)|a|d dν(u)

= K2La|a|−d/2∫V

ψ(2B(u′, v/a)

)Φ(u′) dν(u′)

= K2La|a|−d/2Φ(−v/a),

by the Fourier inversion formula, whereas(t(−1/a) · Φ

)(v) = L−1/a|a|−d/2Φ(−v/a).

Thus the fourth relation is satisfied if and only if K2La = L−1/a for all a ∈ F×. Since

L• is a character of F×, this is equivalent to requiring that La2 = 1 for all a ∈ F× and

K2 = L−1.

Finally, we consider the fifth relation. The approach we used in the finite field case

fails here, because the function u 7→ ψ(bB(u, u)

)(for b ∈ F ) is not integrable over V .

Before we can state the key proposition for this section, we need two results about

squares in F×. We follow the approach suggested in [5] (Chapter 2, §§1.6–7).

Proposition 4.30. Let F be a non-Archimedean local field with odd residue charac-

teristic. Let F×2 denote the subgroup {x2 | x ∈ F×} of F×. Then [F× : F×2] = 4.

Proof. Each x ∈ F× can be written uniquely in the form x = $ky for some k ∈ Zand y ∈ o \ p. So all squares in F× can be written in the form x = $2ky2 for k ∈ Zand y ∈ o \ p, and clearly then y2 + p is a square in (o/p)×. Conversely, if y′ ∈ o \ p

and y′ + p is a square in (o/p)×, then there exists y ∈ o such that y2 = y′ by Hensel’s

lemma (see [8], Proposition 7 on p. 34). It follows that the squares in F× are exactly

those x = $ky such that k is even and y + p is a square in (o/p)×. Since (o/p)× is

cyclic and of even order, its squares form a subgroup of index two.


Corollary 4.31. Maintaining the assumptions of Proposition 4.30, letE be a quadratic

extension of F , and denote by N : E → F the norm map corresponding to this exten-

sion. Then [F× : N(E×)] = 2.

Proof. We can write E = F (√c) for some nonsquare c ∈ F ; then the norm is given by

N(a + b√c) = a2 − b2c. It is then clear that F×2 ≤ N(E×) ≤ F×, so it will suffice to

show that F×2 6= N(E×) and N(E×) 6= F×.

Let γ ∈ o \ p be any element such that γ + p generates the cyclic group (o/p)×;

then γ, $ and $γ are all nonsquare in F×. By Proposition 4.30, they are a complete

set of coset representatives for F×2 in F×. It suffices to consider the cases E = F (√c)

for c = γ, $ and $γ.

If c = $ or $γ, then N(√c) = −c is not a square from F×. In the case where c = γ,

we observe that γ + p is not square in o/p, so there is an extension E = (o/p)(√γ + p)

of o/p. By Lemma 3.19, there exist x, y ∈ o such that x2 − y2γ + p = γ + p, which is

not a square in o/p. In both cases, F×2 6= N(E×).

Suppose c = $ or $γ and suppose a2 − b2c = γ for some a, b ∈ F . Since |b2c| =

q−2k−1 and |a2| = q−2l for some k, l ∈ Z, we must have k ≥ 0. Then a2 + p = γ + p,

which is impossible. Now suppose c = γ and suppose a2 − b2γ = $ for some a, b ∈ F .

Since p = a2 − b2γ + p is a norm from E , we must have a, b ∈ p, which means that

a2 − b2γ ∈ $2o, a contradiction. In both cases, N(E×) 6= F×.

By virtue of Corollary 4.31, for each quadratic extension E of F there exists a

unique nontrivial character of F× which is trivial on N(E×), called the quadratic

character of F× attached to the extension E.

By Lemma 3.14, we can find a basis b1, . . . , bd of V and constants c1, . . . , cd ∈ F×such that B(x1b1 + · · ·+xdbd, y1b1 + · · ·+ydbd) = c1x1y1 + · · ·+ cdxdyd for all xi, yi ∈ F .

Let ∆ = (−1)d/2c1 . . . cd. Let ξ be the trivial character of F× if ∆ is a square in F ,

and the quadratic character of F× attached to the extension F (√

∆) otherwise.

For notational convenience, for all b ∈ F we define Fb : V → C by Fb(v) =

ψ(bB(v, v)

)for all v ∈ V . The action of

(1 b

1

)under η can then be written as η

(1 b

1

)Φ =

FbΦ, where juxtaposition of functions V → C denotes pointwise multiplication.

Proposition 4.32. For all Φ ∈ S(V ) and b ∈ F , the convolution Fb ∗ Φ lies in S(V ).

In addition, there exists a constant δ ∈ {±1,±i} satisfying δ2 = ξ(−1), such that

Fb ∗ Φ = δξ(b)|b|−d/2F−1/bΦ

for all b ∈ F× and Φ ∈ S(V ).

Proof. Let V ′ be a finite-dimensional vector space over F , let B′ be a nondegenerate

symmetric bilinear form on V ′ and let ν ′ be the Haar measure on V ′ which is self-

dual with respect to the pairing P ′ : (u, v) 7→ ψ(2B′(u, v)

). Denote by Φ ∗′ Ψ the

convolution of the functions Φ and Ψ with respect to ν ′, and denote by F ′ the Fourier

transform on S(V ′) with respect to P ′. Let fB′ : V ′ → C be the function given by


fB′(v) = ψ(B′(v, v)

)for all v ∈ V ′. (In particular, Fb = fbB for all b ∈ F×.) Then, for

all Φ ∈ S(V ′) and v ∈ V ′,

(fB′ ∗′ Φ)(v) =

∫V

ψ(B′(v − u, v − u)

)Φ(u) dν ′(u)

= ψ(B′(v, v)

) ∫V

ψ(2B(u,−v) +B(u, u)

)Φ(u) dν ′(u) = fB′(v)F ′(fB′Φ)(−v). (4.8)

Now fB′ and Φ are both locally constant functions of compact support, and F ′ preserves

S(V ′), so fB′ ∗′ Φ ∈ S(V ′).

For each such choice of V ′ and B′ we define a number γ(B′) ∈ T using the projec-

tive representation η′ of SL(2, F ) on L2(V ′), which is defined as in Section 4.4. The

constants used in that representation are unimportant, but we will denote them by K ′

and L′a (for a ∈ F×) to acknowledge that they depend on V ′ and B′. In our presenta-

tion for SL(2, F ) (Proposition 2.15), it is clear from the fourth and fifth relations that

wn(−1)w = n(1)wn(1) and hence that n(−1)w = wn(1)t(−1)wn(1). We will apply

η′ to both sides of this equation and write the action of η′ on S(V ′) by a raised dot.

Since η′ is a unitary projective representation, there exists γ(B′) ∈ T such that

F ′(fB′ ∗′ Φ)1= F ′

(K ′−1L′−1

−1 n(1) · t(−1) · w · n(1) · Φ)

= K ′−2L′−1−1 w · n(1) · w · t(−1) · n(1) · Φ

= γ(B′)K ′−1 n(−1) · w · Φ= γ(B′) f−B′ F ′Φ (4.9)

for all Φ ∈ S(V ′), where step 1 follows from (4.8).

For any b ∈ F×, we can consider the special case where V ′ = V and B′ = bB. We

have F ′(fbB ∗′ Φ) = γ(bB) f−bB F ′Φ for all Φ ∈ S(V ). To determine ν ′, note that∫V

ψ(2bB(u, v)

)(∫V

ψ(2bB(w, u)

)Φ(w) dν(w)

)dν(u)

2= |b|−d

∫V

ψ(2B(u′, v)

)(∫V

ψ(2B(w, u′)

)Φ(w) dν(w)

)dν(u′) = |b|−dΦ(−v)

for all Φ ∈ S(V ) and v ∈ V by the Fourier inversion formula, where we substi-

tuted u′ = bu at step 2. Thus we have ν ′ = |b|d/2ν, since ν ′ is self-dual with re-

spect to P ′. Thus |b|d/2F ′(fbB ∗ Φ) = γ(bB) f−bB F ′Φ for all Φ ∈ S(V ), where ∗denotes convolution with respect to ν. Evaluating this equation at v/b, we obtain

F(fbB ∗ Φ)(v) = |b|−d/2γ(bB) f−b−1B(v)FΦ(v) for all v ∈ V . Let δ = γ(B). It now

suffices to prove that γ(bB) = ξ(b)γ(B) for all b ∈ F×.

We first need some properties of γ. For i = 1, 2, let Vi be a finite-dimensional F -

vector space, Bi a nondegenerate symmetric bilinear form on Vi, νi the corresponding

self-dual Haar measure on Vi and Fi the corresponding Fourier transform on S(Vi).

Define the symmetric bilinear form B′ on V ′ := V1 ⊕ V2 by B′((u1, u2), (v1, v2)

)=

B1(u1, v1) + B2(u2, v2), and let F ′ be the corresponding Fourier transform. It is easy


to check that B′ is nondegenerate and ν ′ = ν1× ν2 is the self-dual Haar measure on V ′

with respect to B′. For any functions Φi : Vi → C (for i = 1, 2), consider the function

Φ1 ⊗ Φ2 : V ′ → C defined by (Φ1 ⊗ Φ2)(v1, v2) = Φ1(v1)Φ2(v2). We easily see that

fB′ = fB1 ⊗ fB2 . By Fubini’s theorem,

F ′(fB′ ∗ (Φ1 ⊗ Φ2)

)= F ′

((fB1 ∗ Φ1)⊗ (fB2 ∗ Φ2)

)= F1(fB1 ∗ Φ1)⊗F2(fB2 ∗ Φ2)

for all Φ1 ∈ S(V1) and Φ2 ∈ S(V2). Therefore, by (4.9),

γ(B′) = γ(B1)γ(B2). (4.10)

Also note that fB′(v) = f−B′(v) = f−B′(−v) and F ′Φ(v) = F ′Φ(−v) for all Φ ∈ S(V ′)

and v ∈ V ′, so taking conjugates of (4.9) shows that γ(B′)−1 = γ(B′) = γ(−B′).We introduce the notation QF(c1, . . . , ck) for a k-dimensional vector space over F ,

together with a symmetric bilinear form which can be diagonalised to the matrix

diag(c1, . . . , ck).

We also introduce the Hilbert symbol (·, ·) on F×. For a, b ∈ F×, we set (a, b) = 1

if there exist x, y, z, w ∈ F , not all zero, such that x2−ay2− bz2 +abw2 = 0; otherwise

we set (a, b) = −1. The Hilbert symbol is clearly symmetric in its arguments. If b is a

square, then (a, b) = 1 for all a ∈ F×. If b is not a square, then (a, b) = 1 if and only if a

is a norm from F (√b), because if x, y, z, w are not all zero then x2−ay2−bz2+abw2 = 0

is equivalent to a = N((x+ z

√b)/(y +w

√b)). As a special case, we see that (a,∆) =

ξ(a) for all a ∈ F×. For all b ∈ F×, a 7→ (a, b) is a character of F×, so the Hilbert

symbol is bilinear. If b = −a then (x, y, z, w) = (0, 1, 1, 0) is a solution, so (a,−a) = 1.

We claim that

γ(QF(1,−a,−b, ab)

)= (a, b) (4.11)

for all a, b ∈ F×. Suppose that (a, b) = 1. If b is a square, then a change of basis gives

QF(1,−a,−b, ab) ∼= QF(1,−a,−1, a), so (4.10) gives


)= γ

(QF(1,−a)

)γ(QF(−1, a)

)= 1.

Otherwise a is a norm from F (√b), so we can find x0, z0 ∈ F , not both zero, such that

a = x20 − bz2

0 . Then

x2 − ay2 − bz2 + cw2 = x2 − bz2 − (x20 − bz2

0)(y2 − bw2)

= x2 − bz2 − (x0y + bz0w)2 + b(x0w + z0y)2,

so a change of basis gives QF(1,−a,−b, ab) ∼= QF(1,−b,−1, b). Thus, once again,


)= γ

(QF(1,−b)

)γ(QF(−1, b)

)= 1.

Suppose now that (a, b) = −1. Consider the quaternion algebra Q, which is

defined to be an F -vector space with basis {1, i, j, k}, subject to the multiplication

rules 1x = x1 = x for x = i, j, k, as well as i2 = a, j2 = b, k2 = −ab, ij = −ji = k,

jk = −kj = −bi, and ki = −ik = −aj. This is an associative algebra with 1. Let


Q have the product topology of four copies of F . If we define the conjugate of an

element x + yi + zj + wk to be x+ yi+ zj + wk = x− yi− zj − wk, it is clear from

the definition that αα = αα = x2− ay2− bz2 + abw2 for all α = x+ yi+ zj +wk ∈ Q.

The value αα can be regarded as an element of F ; it is known as the reduced norm

of α, and is denoted Nrd(α). It is easy to check that Nrd(αβ) = Nrd(α) Nrd(β) for

all α, β ∈ Q. There is an obvious nondegenerate symmetric bilinear form B′ on Q

such that B′(α, α) = Nrd(α); under that bilinear form, Q ∼= QF(1,−a,−b, ab). Since

(a, b) = −1, nonzero elements have nonzero reduced norms, so every nonzero element α

of Q has a left and right inverse Nrd(α)−1α. Thus Q is in fact a division algebra. Let

U = Nrd−1($−mo), where m is an integer that will be specified later, and let

U ′ ={v ∈ Q

∣∣ ψ(2B′(u, v))

= 1 for all u ∈ U}.

Let x0 ∈ F such that ψ(x0) 6= 1. For t = 1, i, j, k we can find at ∈ E such that

2B′(t, at) = x0, since B′ is nondegenerate. Note that 2B′(x1 +xii+xjj+xkk, atx−1t t) =

x0 for all t ∈ {1, i, j, k} and x1, xi, xj, xk ∈ F . Thus if x1 + xii + xjj + xkk ∈ U ′ then

atx−1t t /∈ U for all t ∈ {1, i, j, k}. In particular, a2

tx−2t Nrd(t) = Nrd(atx

−1t t) /∈ $−mo

and hence |xt| < q−m/2|at||Nrd(t)|1/2 for all t ∈ {1, i, j, k}. Therefore we may choose

m large enough that ψ(Nrd(v)

)= 1 for all v ∈ U ′. Now

fB′ ∗′ 1U ′(v) =

∫U ′fB′(v − u) dν ′(u) = fB′(v)

∫U ′ψ(−2B(u, v)

)fB′(u) dν ′(u)

1= fB′(v)

∫U ′ψ(−2B(u, v)

)dν ′(u)

2= ν ′(U ′)fB′(v)1U(v)

for all v ∈ Q, where step 1 follows because fB′(u) = 1 for all u ∈ U ′, and step 2 follows

by definition of U ′. Now evaluating (4.9) at zero with Φ = 1U ′ and dividing by ν ′(U ′)

yields∫UfB′(v) dν ′(v) = γ(B′). This will allow us to calculate γ(B′). It is easy to

check that left-multiplication by x + yi + zj + wk in Q corresponds to multiplication

by a matrix over F whose determinant is (x2 − ay2 − bz2 + abw2)2. It follows that

the measure λ on Q× defined by λ(S) =∫S|Nrd(v)|−2 dν ′(v) is invariant under left-

translation by elements of Q×; in other words, it is a left multiplicative Haar measure

on Q×. So we can calculate that

γ(B′) =

∫U

fB′(v)|Nrd(v)|2 dλ(v)1= C

∫$−mo

ψ(x)|x|2 dµ(x) = C

∫$−mo

ψ(x)|x| dµ(x)

for some positive constant C, where µ is the multiplicative Haar measure on F× given

by µ(S) =∫S|x|−1 dµ(x). Step 1 is justified by the fact that Nrd : Q× → F× is a

group homomorphism and the integrand is constant on the cosets of its kernel in Q×.

We evaluate the above integral by subdividing the domain into subsets on which |x|is constant. Let c be the smallest integer such that ψ($co) = 1. By Lemma 4.18, the

integral In =∫$no

ψ(x) dµ(x) is equal to q−n if n ≥ c and zero otherwise. Thus

∫$no\$n+1o

ψ(x)|x| dµ(x) = |$n|(In − In+1) =

q−n(q−n − q−n−1) if n ≥ c;

q−c+1(−q−c) if n = c− 1; and

0 if n ≤ c− 2.


We may assume m was chosen large enough that −m ≤ c− 1. Then

γ(B′) = C

(−q−2c+1 + (1− q−1)

∞∑n=c

q−2n

)= C

(−q−2c+1 +

q−2c

1 + q−1

)= −C q−2c+1

1 + q−1,

which is negative. Since |γ(B′)| = 1, we conclude that γ(B′) = −1, so (4.11) is proved.

Now (4.11) implies that γ(QF(1, ab)

)= (a, b)γ

(QF(a, b)

)for a, b ∈ F×. To prove

that γ(bB) = (b,∆)γ(B), by writing V as a direct sum of two-dimensional subspaces

it will suffice to prove that γ(QF(bc1, bc2)

)= (b,−c1c2)γ

(QF(c1, c2)

)for b, c1, c2 ∈ F×.

Indeed,

γ(QF(bc1, bc2)

)= (bc1, bc2)γ

(QF(1, b2c1c2)

)= (b,−b)(b,−c1)(b, c2)(c1, c2)γ

(QF(1, c1c2)

)= (b,−c1c2)γ

(QF(c1, c2)

).

Finally, we must prove that δ2 = ξ(−1), or in other words that γ(B)2 = (∆,−1).

Again decomposing V into two-dimensional subspaces, it will suffice to prove that

γ(QF(c1, c2)

)2= (−c1c2,−1). By (4.11),

γ(QF(c1, c2)

)2= γ

(QF(1, c1c2)

)2= γ

(QF(1, 1, c1c2, c1c2)

)= (−1,−c1c2).

If we make the definitions La := ξ(a) and K := δ, then our necessary and sufficient

criteria for the first four relations to hold, namely that K2 = L−1 and La2 = 1 for all

a ∈ F×, are satisfied. Furthermore, the fifth relation holds with these definitions, as

we can readily verify, for any Φ ∈ S(V ), v ∈ V and b ∈ F×:(w · n(b) · w · Φ

)(v) =

(w · (δFbΦ)

)(v)

=(w · ξ(−b)|b|−d/2(F−1/b ∗ Φ))(v)

= δξ(−b)|b|−d/2(F−1/b ∗ Φ)(−v),

by Proposition 4.32 and the Fourier inversion formula, whereas(t(−1/b) · n(−b) · w · n(−1/b) · Φ

)(v)

= ξ(−b)|b|−d/2F−b(−v/b)δF−1/bΦ(−v/b)

= δξ(−b)|b|−d/2ψ(−B(v, v)/b

) ∫V

ψ(2B(u,−v/b)−B(u, u)/b

)Φ(u) dν(u)

= δξ(−b)|b|−d/2∫V

ψ(−B(−v − u,−v − u)/b

)Φ(u) dν(u)

= δξ(−b)|b|−d/2(F−1/b ∗ Φ)(−v).

Using the fact that t(−1/b) · n(−b) · w · n(−1/b) · Φ = n(−1/b) · w · n(−b) · t(−b) · Φfor all b ∈ F× and Φ ∈ S(V ), which follows from the earlier relations, we conclude

that the fifth relation holds. So the map η specified above on generators of SL(2, F )

extends to a unique representation of SL(2, F ) on L2(V ).


4.6 Extending the representation to GL(2, F )

By Proposition 4.30, the norm map of a quadratic extension of F is not surjective.

Thus is not possible to construct a representation of GL(2, F ) directly in the same way

as in Section 3.5. We instead extend the representation η obtained in the previous

section to the subgroup of GL(2, F ) consisting of matrices whose determinants are

norms, and then use the induced representation.

Let E be a quadratic extension of F . Denote by a 7→ a the nontrivial Galois

automorphism of E fixing F , and denote the corresponding trace and norm maps by

tr(x) = x+ x and N(x) = xx. Let E×1 be the set of elements of E of norm 1. Let χ be

a quasicharacter of E× such that χ(E×1 ) 6= 1. We specialise η to the case where V = E

and B(u, v) = 12

tr(uv). The corresponding Fourier transform Φ of Φ ∈ S(E) is given

by Φ(v) =∫Eψ(tr(uv)

)Φ(u) dν(u) for all v ∈ E, where ν is the self-dual measure on

E with respect to the pairing (u, v) 7→ ψ(tr(uv)

).

We can write E = F (√

∆′) for some nonsquare ∆′ ∈ F ; then N(a + b√

∆′) =

a2 − b2∆′, so in the terminology of Section 4.5 we have ∆ = −1(−∆′) = ∆′. It follows

that ξ (as defined in that section) is in fact the quadratic character of F× attached to

the extension E.

Let GL(2, F )+ be the subgroup of GL(2, F ) (of index two) consisting of all the

matrices whose determinants lie in N(E×).

Consider the subspace Uχ of S(E) defined by

Uχ ={

Φ ∈ S(E)∣∣ Φ(tx) = χ(t)−1Φ(x) for all t ∈ E×1

}.

Theorem 4.33. There exists an admissible representation θχ of GL(2, F )+ on Uχ such

that, for all Φ ∈ Uχ and v ∈ E,(θχ

(1 b

1

)Φ

)(v) = ψ

(bN(v)

)Φ(v) for b ∈ F ,(

θχ

(a

1/a

)Φ

)(v) = |a|ξ(a)Φ(av) for a ∈ F×,(

θχ

(a

1

)Φ

)(v) = |a|1/2χ(c)Φ(cv)

for a ∈ N(E×),

where N(c) = a,

and θχ

(1

−1

)Φ = δΦ,

where δ ∈ T is the constant determined by Proposition 4.32. The representation

θGL(2,F )χ of GL(2, F ) induced from θχ is also admissible.

Proof. By essentially the same calculations as in the finite field case, Uχ is invariant

under the action of η.

We apply Lemma 3.20 once again to determine whether the representations of

SL(2, F ) and P :={(

a1

) ∣∣ a ∈ N(E×)}

on Uχ are compatible. If we once again

4.6. Extending the representation to GL(2, F ) 59

denote the matrix(a

1

)by c(a), and denote the actions of SL(2, F ) and P both by a

raised dot, then we easily calculate that, if N(c) = a,(s(a) · n(b) · s(1/a) · Φ

)(v) = |a|1/2χ(c)ψ

(bN(cv)

)(s(1/a) · Φ

)(cv)

= |a|1/2χ(c)ψ(abN(v)

)|a|−1/2χ(1/c)Φ(v) =

(n(ab) · Φ

)(v);(

s(a) · t(A) · s(1/a) · Φ)(v) = |a|1/2χ(c)|A|ξ(A)|a|−1/2χ(1/c)Φ(cAc−1v)

=(t(A) · Φ

)(v); and(

s(a) · w · s(1/a) · Φ)(v) = |a|1/2χ(c) δ

∫E

ψ(tr(ucv)

)|a|−1/2χ(1/c)Φ(u/c) dν(u)

1= δ

∫E

ψ(tr(u′av)

)Φ(u′) |N(c)|dν(u′)

= |a|ξ(a) δ

∫E

ψ(tr(u′av)

)Φ(u′) dν(u′) =

(t(a) · w · Φ

)(v),

where we substituted u = cu′ at the step labelled 1. The constant obtained in this

step requires some explanation. When we substitute u = cu′ we are performing a

linear transformation of the F -vector space E. We can write c = C + D√

∆ for some

C,D ∈ F , not both zero. The matrix of the transformation u′ 7→ cu′ with respect to

the basis (1,√

∆) of E is then(C D∆D C

), which has determinant C2−D2∆ = N(c). Thus

the integral transforms according to the rule dν(cu′) = |N(c)|dν(u′). By Lemma 3.20,

there exists a unique representation θχ satisfying the stated equations.

We now check that θχ is smooth. Let Φ ∈ Uχ. It will suffice to find a neighbourhood

U of the identity matrix in GL(2, F ) such that θχ(g)Φ = Φ for all g ∈ U , because then

US ⊆ S, where S denotes the stabiliser of Φ in GL(2, F ), and this proves that S is

open. This prompts us to consider the matrix decomposition(a b

c d

)=

(1d

d

)(1 bd

1

)(ad− bc

1

)(−1

−1

)(1

−1

)(1 − c

d

1

)(1

−1

),

(4.12)

which holds for all(a bc d

)∈ GL(2, F ) with d 6= 0. Since Φ ∈ S(E), supp Φ is bounded,

and hence there exists k ∈ N such that ψ(− cdN(v)

)= 1 whenever c ∈ $ko, d ∈ o

and v ∈ supp Φ, since ψ is locally constant. Thus for all c ∈ $ko and d ∈ o we have

F(F−c/dΦ

)= F

(Φ), and hence

t(−1) · w · n(−c/d) · w · Φ = δ2ξ(−1)Φ = Φ (4.13)

by the Fourier inversion formula.

Since χ is locally constant, there exists k′ ∈ N such that χ(c′) = 1 whenever

N(c′) ∈ 1 + $k′o. In addition, since Φ is locally constant and of compact support,

there exists a finite set I and constants xi ∈ E and ni ∈ N for i ∈ I such that Φ

is constant on each of the sets Ui = {x ∈ E | N(x − xi) ∈ $nio} and such that

supp Φ ⊆⋃i∈I Ui. Let m be an integer such that N(supp Φ) ⊆ $mo. Now if k′′ ∈ N

and c′ ∈ 1 + $k′′o then N(c′v − v) = N(v)N(c′ − 1) ∈ $m+k′′o for all v ∈ supp Φ.

Setting k′′ = maxi∈I ni −m, we see that if c ∈ 1 +$k′′o and v ∈ Ui then

|N(c′v − xi)| ≤ max{|N(c′v − v)|, |N(v − xi)|} ≤ q−ni


by the ultrametric inequality, and therefore Φ(c′v) = Φ(xi) = Φ(v). Finally, there

exists k′′′ ∈ N such that ψ(bdN(v)

)= 1 whenever b ∈ $k′′o and d ∈ o.

Let n = max{k, k′, k′′, k′′′}. Now suppose a, d ∈ 1 + $no and b, c ∈ $no. We

have |ad − bc| = 1, |1/d| = 1, ad − bc ∈ 1 + $no and N(a) = a2 ∈ 1 + $no. Also,

1/d+o = 1+o, so by Hensel’s lemma 1/d is a square in F×, so ξ(1/d) = 1. Putting all

the above facts together, we conclude that(t(1/d) ·n(bd) · s(ad− bc) ·Φ

)(v) = Φ(v) for

all v ∈ supp Φ. Combining this with (4.12) and (4.13), we conclude that(a bc d

)·Φ = Φ

for all(a bc d

)∈ GL(2, F ) such that

∥∥( a bc d

)− I∥∥∞ ≤ q−n. Thus θχ is smooth.

Now we show that θχ is admissible. Let H be an open subgroup of GL(2, F ), and

let W = {Φ ∈ Uχ | θχ(h)Φ = Φ for all h ∈ H}. Since ψ is nontrivial and locally

constant, we can fix some x0 ∈ F such that ψ(x0) 6= 1 and some nonzero x1 ∈ F

such that ψ(x) = 1 whenever |x| < |x1|. Since H is open, there exists n ∈ N such

that(a bc d

)∈ H whenever a, d ∈ 1 + $no and b, c ∈ $no. Pick any Φ ∈ W . On the

one hand, n(b) · Φ = θχ(

1 b1

)Φ = Φ for all b ∈ $no, which means that ψ

(bN(v)

)= 1

for all v ∈ supp Φ and b ∈ $no. Since ψ(x0) 6= 1, it follows that N(v) 6= x0/b for

all v ∈ supp Φ and b ∈ $no, and so |N(v)| < qn|x0| for all v ∈ supp Φ. Thus there

is a closed and bounded set K ⊆ E, independent of Φ, such that supp Φ ⊆ K for

all Φ ∈ W . On the other hand, t(−1) · w · n(b) · w · Φ = θχ(

1−b 1

)Φ = Φ for all

b ∈ $no. Acting on both sides by w, we see that n(b) · w · Φ = w · Φ for all b ∈ $no,

which shows that supp Φ ⊆ K for all Φ ∈ W , by the same argument that showed

supp Φ ⊆ K. Note that if u = x + y√

∆ and v = z + w√

∆ then tr(uv) = xz − yw∆,

and so |tr(uv)| ≤ max{|x||z|, |y||w||∆|}. Since K is bounded, we can find an open

neighbourhood U of 0 in E such that | tr(uv)| ≤ |x1| whenever u ∈ K and v ∈ U , so in

particular ψ(tr(uv)

)= 1 whenever u ∈ K and v ∈ U . Now, for all Φ ∈ W , the Fourier

inversion formula implies that

Φ(−v − v′) =

∫K

ψ(tr(uv)

)ψ(tr(uv′)

)Φ(u) dν(u) =

∫K

ψ(tr(uv)

)Φ(u) dν(u) = Φ(−v)

for all Φ ∈ W , v ∈ E and v′ ∈ U . For each v ∈ K let Ev = {u ∈ E | u − v ∈ U};these sets are open and cover K. Since K is closed and bounded, it is a closed subset

of the compact set $k0o for some k0 ∈ Z, so K is compact. Thus we can find a finite

set K ′ ⊆ K such that the Ev for v ∈ K ′ cover K. But, for all Φ ∈ W , Φ is constant

on each Ev and has support in K, so Φ is determined by the values Φ(v) for v in the

finite set K ′. Thus dimW <∞. We have now proved that θχ is admissible.

Finally, it is easy to check, from the definition of the induced representation, that

the representation induced from an admissible representation of an index two subgroup

is itself admissible. Thus θGL(2,F )χ is admissible.

The representation θGL(2,F )χ is called a dihedral representation. With further work, it

can be shown that the dihedral representations have a property called supercuspidality

analogous to Definition 3.24, and that they are irreducible—see [1], Theorem 4.8.6

on pp. 541–2. These representations are remarkably close analogues of the cuspidal

representations θ(χ) discussed in Section 3.6.

Bibliography

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Cambridge (1997).

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[4] G. B. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton

(1995).

[5] I. M. Gel’fand, M. I. Graev and I. I. Pyatetskii–Shapiro, Representation Theory

and Automorphic Functions, W. B. Saunders, Philadelphia (1969).

[6] M. Hall, The Theory of Groups, Macmillan, New York (1959).

[7] I. Piatetski-Shapiro, Complex Representations of GL(2, K) for Finite Fields K,

Contemporary Mathematics 16, The American Mathematical Society, Providence,

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[8] J.-P. Serre, Local Fields, Springer-Verlag, New York (1979).

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