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
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