10
ALESSANDRO FIGA-TALAMANCA
of a step one translation on (0, oo). This means that
Goo
is generated by
Boo
and
a step one translation along the geodesic (0, oo).
We should observe at this point that a step one translation on the geodesic (0, oo)
is a dilation which may be realized through multiplication by p.
THE AFFINE GROUP OFF
We let
Af f(F)
be the group of affine transformation
ofF
onto itself, that is the
group of transformations
a --. aa
+
(3 with a, (3 E
F
and a
=/=-
0. It is easy to show
that
Af f(F)
is the semidirect product of the group of dilations
a --. aa,
which is
isomorphic to the multiplicative group of nonzero elements of F and the group of
translations
a--. a+
(3, which is isomorphic to the additive group
ofF,
the group
of translations being normal in
Af f (F).
We observe that translations are indeed isometries of F. They can be identified
therefore with automorphisms of
X
(the tree of
F)
which leave horocycles invariant.
Multiplication by a non zero element
a
E
F is an isometry only if
laiF
=
1. Thus
Af f(F)
n
Boo
consists of the affine transformations
a --. aa
+
(3 with
Ia lF
=
1.
In analogy to the Euclidean motion group of the complex plane, which is similarly
generated by translations and multiplications by complex numbers of modulus one,
we shall call
Af f(F)
n
Boo
the Euclidean motion group of
F.
We denote this group
by
M(F).
It is clear that
M(F)
is isomorphic to the semidirect product
ox
1
F
of
ox
and the additive group of (translations by elements of) F.
Finally we observe that
M(F)
is open and normal in
Af f(F)
and that the
quotient group
Af f (F)
I
M (F)
is isomorphic to Z. In this case too
Af f (F)
is
generated by
M(F)
and multiplication by p.
SPHERICAL REPRESENTATIONS OF THE GROUPS
I
som(F)
AND
M(F)
The representation theory of the group
M(F)
is well known [4]. Since
M(F)
is
isomorphic to
ox
1
F,
its irreducible unitary representations may be obtained by
trivially extending to the group
M(F)
the characters of
ox
and inducing to
M(F)
the characters of the additive group F.
The representation theory of I
som(F)
is also known, because it is the same as
the representation theory of
Boo,
which is described in [5]. We shall give here a more
explicit treatment of the main property of unitary representations of I
som(F) '::::'.
B
00
:
all irreducible representations except the identity have coefficients of compact
support.
Up to a point we shall give a unified treatment to both groups. We can thus
determine all the so-called
spherical
representations. Let
G
be a closed subgroup
of
I som( F)
with the property that given
a, a', b, b'
E
F
such that
d( a', b')
=
d( a, b),
there exists
g
E
G
such that
ga
=
a'
and
gb
=
b'.
In other words we suppose that
the action of
G
on
F
is
doubly transitive.
Observe that G may well be
I som(F)
or
M(F).
We consider now the stabilizer in
G
of an element of
F.
For convenience we
choose this element to be 0. Thus we consider
Ko
=
{g
E
G: gO=
0}.
The coset space
G
I
K
0
may be identified canonically with the space
F,
through
the map
g --. gO,
which is constant on the coset
gK0
.
Observe that in the same
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