LOCAL FIELDS AND TREES 13

other words

f(a)

=

0 if

laiF

q-n+l. Iff

is zero on

{a

E

F:

laiF:::;

q-n},

then

En-d

=

0 implies that f must be zero also on

{a

E F :

Ia

lF

=

q-n+l} where it is

constant.

Iff

is not zero on

{a

E

F :

Ia

lF :::;

q-n}

we may assume that

f

takes the

value one. Then

En-d=

0 implies that the value off on {a E

F:

laiF

= q-n+l}

must be l~q.

We should notice that the K

0

-invariant functions

Pn

may be interpreted as

K

0-

bi-invariant functions on

G.

As such they are coefficients of representations

7rn.

In

other words

/Jn(g)

=

(q-

1)qn-l

l

/Jn(gh)/Jn(h) dh

This means that

¢n

is the positive definite spherical functions defined by the spher-

ical representation

'lrn·

We prove now that the only other spherical representation

is the trivial representation.

Corollary.

The functions Pn together with the function identically one are the

only spherical functions of the Gelfand pair (G, Ko).

Proof. .

Let 'ljJ be a spherical function on

G.

Since the functions

Pn

(viewed as

functions on G) are continuous K

0

-bi-invariant functions of compact support, by

the Proposition above,

Suppose first that

Pn

*

'ljJ

=

0 for every

n.

Then, because of the special form of ¢n,

we may conclude that 'ljJ has the same integral over every ball of radius

q-n.

Since

'lj;(1c)

=

1

this means that 'ljJ is indentically one. On the other hand if

¢n *'l/J(1c)

-1-

0

then for m

-1-

n,

0 = ¢m

*

'1/J,

because ¢m

*

Pn

= 0. Thus if 'ljJ is not identically one,

Pn * 'lj;(1c)

-1-

0 for exactly on

n

and, for the same

n,

1

'1/J

=

Pn *'l/J(1c) '1/J*Pn·

Since the operators

Dn

(which are defined for continuous functions as well as for

square integrable functions) commute with convolution and

Dn¢n

=

¢n, we con-

clude that

Dn'l/J

=

'1/J.

We argue now as in the proof of the Lemma to prove that 'lj;

is a multiple of ¢n, and indeed it equals

Pn

because it is one at the identity.

NONSPHERICAL REPRESENTATIONS OF THE GROUPS

Isom(F)

AND

M(F)

It is not difficul.t to show that the irreducible representations of M(F) which are

not spherical are obtained by trivially extending to all of

M(F)

the characters of

()X,

The situation is completely different for the group

I som(

F). In order to study

the nonspherical irreducible representations we must consider the action of

I som(

F)

on the tree ofF, in other words we consider the isomorphic realization of

I som(F)

as the subgroup

Boo

which fixes the boundary point oo and stabilizes the horocycles

with respect to the same point. From this point of view the irreducible representa-

tions of

Boo

and consequently of

I som(

F) are studied in [5]. In these lectures we

will prove only that, except for the trivial representation, they all have a coefficient

of compact support.

We introduce now a special kind of finite subtree of a homogeneous tree

X.