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.

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.