10

ALESSANDRO FIGA-TALAMANCA

of a step one translation on (0, oo). This means that

Goo

is generated by

Boo

and

a step one translation along the geodesic (0, oo).

We should observe at this point that a step one translation on the geodesic (0, oo)

is a dilation which may be realized through multiplication by p.

THE AFFINE GROUP OFF

We let

Af f(F)

be the group of affine transformation

ofF

onto itself, that is the

group of transformations

a --. aa

+

(3 with a, (3 E

F

and a

=/=-

0. It is easy to show

that

Af f(F)

is the semidirect product of the group of dilations

a --. aa,

which is

isomorphic to the multiplicative group of nonzero elements of F and the group of

translations

a--. a+

(3, which is isomorphic to the additive group

ofF,

the group

of translations being normal in

Af f (F).

We observe that translations are indeed isometries of F. They can be identified

therefore with automorphisms of

X

(the tree of

F)

which leave horocycles invariant.

Multiplication by a non zero element

a

E

F is an isometry only if

laiF

=

1. Thus

Af f(F)

n

Boo

consists of the affine transformations

a --. aa

+

(3 with

Ia lF

=

1.

In analogy to the Euclidean motion group of the complex plane, which is similarly

generated by translations and multiplications by complex numbers of modulus one,

we shall call

Af f(F)

n

Boo

the Euclidean motion group of

F.

We denote this group

by

M(F).

It is clear that

M(F)

is isomorphic to the semidirect product

ox

1

F

of

ox

and the additive group of (translations by elements of) F.

Finally we observe that

M(F)

is open and normal in

Af f(F)

and that the

quotient group

Af f (F)

I

M (F)

is isomorphic to Z. In this case too

Af f (F)

is

generated by

M(F)

and multiplication by p.

SPHERICAL REPRESENTATIONS OF THE GROUPS

I

som(F)

AND

M(F)

The representation theory of the group

M(F)

is well known [4]. Since

M(F)

is

isomorphic to

ox

1

F,

its irreducible unitary representations may be obtained by

trivially extending to the group

M(F)

the characters of

ox

and inducing to

M(F)

the characters of the additive group F.

The representation theory of I

som(F)

is also known, because it is the same as

the representation theory of

Boo,

which is described in [5]. We shall give here a more

explicit treatment of the main property of unitary representations of I

som(F) '::::'.

B

00

:

all irreducible representations except the identity have coefficients of compact

support.

Up to a point we shall give a unified treatment to both groups. We can thus

determine all the so-called

spherical

representations. Let

G

be a closed subgroup

of

I som( F)

with the property that given

a, a', b, b'

E

F

such that

d( a', b')

=

d( a, b),

there exists

g

E

G

such that

ga

=

a'

and

gb

=

b'.

In other words we suppose that

the action of

G

on

F

is

doubly transitive.

Observe that G may well be

I som(F)

or

M(F).

We consider now the stabilizer in

G

of an element of

F.

For convenience we

choose this element to be 0. Thus we consider

Ko

=

{g

E

G: gO=

0}.

The coset space

G

I

K

0

may be identified canonically with the space

F,

through

the map

g --. gO,

which is constant on the coset

gK0

.

Observe that in the same

ALESSANDRO FIGA-TALAMANCA

of a step one translation on (0, oo). This means that

Goo

is generated by

Boo

and

a step one translation along the geodesic (0, oo).

We should observe at this point that a step one translation on the geodesic (0, oo)

is a dilation which may be realized through multiplication by p.

THE AFFINE GROUP OFF

We let

Af f(F)

be the group of affine transformation

ofF

onto itself, that is the

group of transformations

a --. aa

+

(3 with a, (3 E

F

and a

=/=-

0. It is easy to show

that

Af f(F)

is the semidirect product of the group of dilations

a --. aa,

which is

isomorphic to the multiplicative group of nonzero elements of F and the group of

translations

a--. a+

(3, which is isomorphic to the additive group

ofF,

the group

of translations being normal in

Af f (F).

We observe that translations are indeed isometries of F. They can be identified

therefore with automorphisms of

X

(the tree of

F)

which leave horocycles invariant.

Multiplication by a non zero element

a

E

F is an isometry only if

laiF

=

1. Thus

Af f(F)

n

Boo

consists of the affine transformations

a --. aa

+

(3 with

Ia lF

=

1.

In analogy to the Euclidean motion group of the complex plane, which is similarly

generated by translations and multiplications by complex numbers of modulus one,

we shall call

Af f(F)

n

Boo

the Euclidean motion group of

F.

We denote this group

by

M(F).

It is clear that

M(F)

is isomorphic to the semidirect product

ox

1

F

of

ox

and the additive group of (translations by elements of) F.

Finally we observe that

M(F)

is open and normal in

Af f(F)

and that the

quotient group

Af f (F)

I

M (F)

is isomorphic to Z. In this case too

Af f (F)

is

generated by

M(F)

and multiplication by p.

SPHERICAL REPRESENTATIONS OF THE GROUPS

I

som(F)

AND

M(F)

The representation theory of the group

M(F)

is well known [4]. Since

M(F)

is

isomorphic to

ox

1

F,

its irreducible unitary representations may be obtained by

trivially extending to the group

M(F)

the characters of

ox

and inducing to

M(F)

the characters of the additive group F.

The representation theory of I

som(F)

is also known, because it is the same as

the representation theory of

Boo,

which is described in [5]. We shall give here a more

explicit treatment of the main property of unitary representations of I

som(F) '::::'.

B

00

:

all irreducible representations except the identity have coefficients of compact

support.

Up to a point we shall give a unified treatment to both groups. We can thus

determine all the so-called

spherical

representations. Let

G

be a closed subgroup

of

I som( F)

with the property that given

a, a', b, b'

E

F

such that

d( a', b')

=

d( a, b),

there exists

g

E

G

such that

ga

=

a'

and

gb

=

b'.

In other words we suppose that

the action of

G

on

F

is

doubly transitive.

Observe that G may well be

I som(F)

or

M(F).

We consider now the stabilizer in

G

of an element of

F.

For convenience we

choose this element to be 0. Thus we consider

Ko

=

{g

E

G: gO=

0}.

The coset space

G

I

K

0

may be identified canonically with the space

F,

through

the map

g --. gO,

which is constant on the coset

gK0

.

Observe that in the same