LOCAL FIELDS AND TREES
11
fashion double cosets
KogKo
may be identified with the orbits of
Ko
on
F.
These
orbits, because of the double transitive action of G, are exactly the set of constancy
of the modular function I·IF that is the sets {0} and {a E F: fa[F
=
qn},
for n E
Z.
We consider now the convolution algebras
L
1
(G/Ko)
and
L
1
(Ko\G/Ko),
of the
K
0
-invariant and respectively of the K0 -bi-invariant elements of L1 (G).
Our remarks above imply that L
1
(G/Ko)
may be identified with L
1
(F) and
L
1
(K0 \G/ K0
)
may be identified with the space of integrable functions on F which
are constant on the sets {a
E
F : [a[F
=
qn}
for n
E
Z.
A consequence of the doubly transitive action of
G
on F is that L
1 (K0 \G/K0
)
is a commutative algebra, in other words, according to current terminology that
the pair (G,
Ko)
is a
Gelfand pair.
We shall not review here the theory of Gelfand
pairs. A good reference for the theory of Gelfand pairs is [1], or [3].
We recall orily that if
(G,Ko)
is a Gelfand pair there exists a series of unitary
irreducible representations, called the
spherical representations
with the property of
having a nonzero K0 -invariant vector, that is a vector which is invariant under the
action of the group
Ko
relative to the representation. Spherical representations are
exactly the representations which occur in the decomposition of the unitary action
of
G
on the Hilbert space
L 2 (GjK0
)
as a direct integral (or sum) of irreducible
representations.
Let
1r
be a spherical representation of G. Let
'H1r
be the Hilbert space on which
1r
acts. Let
e
E
'H1r
be a Ko-invariant vector of norm one. Then the function
( 1r(g
)e, e)
is called the
spherical
function associated to
1r.
Positive definite spherical
functions identify uniquely spherical representations. They also define all the pos-
itive multiplicative linear functionals on the commutative algebra
L 1 (K0 \GfK0
).
This is a consequence of the following characterization of (not necessarily positive
definite) spherical functions that we state without proof [3,Chapter IV].
Proposition.
A nonzero continuous Ko -bi-invariant function
¢
is a spherical func-
tion if one of the following equivalent condition holds:
(1)
¢(1c)
= 1 and
¢
is an eigenfunction of right convolution by continuous
Ko-bi-invariant functions of compact support, i.e.
(2)
¢*
f(g) =
¢*
f(1c)¢(g)
for every continuous Ko-bi-invariant function of compact support f.
r
¢(gkg') dx =
¢(g)¢(g'),
}Ko
for every g,g'
E
G.
(3) the map
L:
f
~
Ia
f(g)¢(g) dg
is
a homomorphism of the convolution algebra of all continuous K
0
-bi-
invariant functions of compact support into the algebra
C
of complex num-
bers.
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