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.
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