Contemporary Mathematics

Volume 206, 1997

LOCAL FIELDS AND TREES

ALESSANDRO

FIGA-TALAMANCA

Universita di Roma "La Sapienza"

INTRODUCTION

A local field is a topological field that is locally compact, second countable,

nondiscrete, and totally disconnected. We refer to the paper Local Fields and

Buildings by Tim Steger

[7]

in this volume for a thorough discussion of local fields

and a description of the most important examples.

A

homogeneous tree is naturally associated to a local field

F

much in the same

way the interior of the upper half plane is associated to the field of real numbers.

In other words the boundary of the tree is, in a natural fashion, the one point

compactification of F.

The tree associated to

F

is a very particular instance of a Bruhat-Tits building

and its construction may follow with the appropriate simplification the construction

of a building starting with lattices with respect to F (in this case two-dimensional

lattices) as in [7].

In these lectures we shall take a different and more elementary approach to

the construction of the tree associated to a local field. This approach allows us

to identify in a straightforward fashion certain groups of transformations of

F

as

groups of automorphisms of the associated tree. In particular we will consider the

group of all isometries of F, which we indicate by I som( F). The identification of

I som( F) with a subgroup of the group of automorphism of the tree of F, allows

us to subsume the theory of unitary representation of the latter group, as studied

by C. Nebbia in [5], following the methods introduced by G.I. Ol'shianskii [6] to

classify the unitary representations of the full group of automorphisms. We show

in this fashion that all irreducible unitary representations (other than the identity)

of I som(F) have a coefficient, and hence a dense set of coefficients with compact

support. This property of the unitary representations of Isom(F) may be proved

directly for the so called spherical representations. However consideration of the

tree of

F

seems to be an essential part of the proof for the other representations.

We shall start now with a review of basic facts about homogeneous trees.

HOMOGENEOUS TREES

A tree is a connected graph without circuits. More explicitly, in order to have a

tree we must have first of all a nonempty set X (the set of vertices) and a collection

of two-elements subsets IE (the set of edges). Vertices and edges are enough to

1991

Mathematics Subject Classification.

Primary 22D10, 22E35; Secondary 43A90.

3

http://dx.doi.org/10.1090/conm/206/02684

Volume 206, 1997

LOCAL FIELDS AND TREES

ALESSANDRO

FIGA-TALAMANCA

Universita di Roma "La Sapienza"

INTRODUCTION

A local field is a topological field that is locally compact, second countable,

nondiscrete, and totally disconnected. We refer to the paper Local Fields and

Buildings by Tim Steger

[7]

in this volume for a thorough discussion of local fields

and a description of the most important examples.

A

homogeneous tree is naturally associated to a local field

F

much in the same

way the interior of the upper half plane is associated to the field of real numbers.

In other words the boundary of the tree is, in a natural fashion, the one point

compactification of F.

The tree associated to

F

is a very particular instance of a Bruhat-Tits building

and its construction may follow with the appropriate simplification the construction

of a building starting with lattices with respect to F (in this case two-dimensional

lattices) as in [7].

In these lectures we shall take a different and more elementary approach to

the construction of the tree associated to a local field. This approach allows us

to identify in a straightforward fashion certain groups of transformations of

F

as

groups of automorphisms of the associated tree. In particular we will consider the

group of all isometries of F, which we indicate by I som( F). The identification of

I som( F) with a subgroup of the group of automorphism of the tree of F, allows

us to subsume the theory of unitary representation of the latter group, as studied

by C. Nebbia in [5], following the methods introduced by G.I. Ol'shianskii [6] to

classify the unitary representations of the full group of automorphisms. We show

in this fashion that all irreducible unitary representations (other than the identity)

of I som(F) have a coefficient, and hence a dense set of coefficients with compact

support. This property of the unitary representations of Isom(F) may be proved

directly for the so called spherical representations. However consideration of the

tree of

F

seems to be an essential part of the proof for the other representations.

We shall start now with a review of basic facts about homogeneous trees.

HOMOGENEOUS TREES

A tree is a connected graph without circuits. More explicitly, in order to have a

tree we must have first of all a nonempty set X (the set of vertices) and a collection

of two-elements subsets IE (the set of edges). Vertices and edges are enough to

1991

Mathematics Subject Classification.

Primary 22D10, 22E35; Secondary 43A90.

3

http://dx.doi.org/10.1090/conm/206/02684