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
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