12
ALESSANDRO FIGA-TALAMANCA
We now proceed to identify all spherical representations of the group G. We
consider the left action of G on L
2
(F): ..(g)f(a)
=
f(g- 1a). This action defines a
unitary representation ... We shall presently decompose .. into the sum of spherical
representations.
Consider the partition of F into closed balls of radius q-n. Define En to be the
conditional expectation with respect to this partition.In other words, for
f
E
L2 (F),
let
Enf(a)
=
qn { f(t) dt,
JB(a,q-n)
where B(a, q-n) is the ball of radius q-n which contains a. Let Dn =En-
En-1,
and let Hn
=
DnL2
.
It is easy to see that
It is also immediate that since ..(g)En
=
En..(g) for every g E G, the spaces 1tn are
invariant under ...
We shall prove that the restrictions of.. to 1tn which we shall denote by 7rn,
together with the trivial representation, are exactly the spherical representations
of
G
that is the irreducible unitary representations having a nonzero K
0
-invariant
vector.
We shall use the fact that, since (
G, K
0
)
is a Gelfand pair, a unitary represen-
tation with a nonzero K
0
-invariant vector is irreducible if and only if the space of
K
0
-invariant vectors is one dimensional. Thus irreducibility of 7rn will follow if we
show that the set of K
0
-invariant vectors of Hn is one dimensional.
Observe that 1tn contains a nonzero K
0
-invariant element, namely the function
Indeed
En¢n
=
cPn and En-lcPn
=
0. The latter statement is true because the
set {a:
laiF
=
q-n+l} is the union of exactly q
-1
balls of radius q-n. Thus
Dn¢n
=
cPn and¢
E
1tn.
Lemma. Every K
0
-invariant element of1tn is a multiple of cPn· Consequently the
representations
1r
n are irreducible and spherical.
Proof. Suppose that f
E
Hn and f(ka)
=
f(a) for every k
E
Ka. Then Enf
=
f
implies that
f
is constant on the closed balls of radius q-n. But
f
is also constant
on the orbits of K
0
,
that is on the sets
For k
-n
+
1 these sets are unions of a finite number of closed balls of radius
q-n+I. But En-d
=
0 and hence the average off on any closed ball of radius
q-n+l is zero. We conclude that f is zero on {a
E
F:
Ia lF
=
qk} if k -n
+
1. In
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