Harmonic Analysis and Representation Theory for Groups Acting on Homogenous Trees (London Mathematical Society Lecture Note Series)

These notes treat in full detail the theory of representations of the group of automorphisms of a homogeneous tree. The unitary irreducible representations are classified in three types: a continuous series of spherical representations; two special representations; and a countable series of cuspidal representations as defined by G.I. Ol'shiankii. Several notable subgroups of the full automorphism group are also considered. The theory of spherical functions as eigenvalues of a Laplace (or Hecke) operator on the tree is used to introduce spherical representations and their restrictions to discrete subgroups. This will be an excellent companion for all researchers into harmonic analysis or representation theory. CONTENTS......Page 5 Preface......Page 7 1) Graphs and trees......Page 11 2) The free group as a tree......Page 15 3) Automorphisms of a tree......Page 16 4) The group of automorphisms Aut(X)......Page 20 5) Compact maximal subgroups......Page 22 6) Discrete subgroups......Page 24 7) Cayley graphs which are trees......Page 26 8) Amenable subgroups......Page 28 9) Orbits of amenable subgroups......Page 34 10) Groups with transitive action on the boundary......Page 36 11) Notes and remarks......Page 41 1) Eigenfunctions of the Laplace operator......Page 44 2) Spherical functions......Page 51 3) Intertwining operators......Page 54 4) The Gelfand pair (G,K)......Page 56 5) Spherical representations......Page 60 6) The resolvent of the Laplace operator and the spherical Plancherel formula......Page 66 7) The restriction problem......Page 73 8) Construction and boundedness of P......Page 76 9) Approximating the projection P......Page 78 10) The constant 1 is a cyclic vector......Page 84 11) Notes and remarks......Page 90 1) A classification of unitary representations......Page 94 2) Special representations......Page 97 3) Cuspidal represent at ions and the Plancherel formula of AutU)......Page 108 4) Notes and remarks......Page 124 1) p-adic fields......Page 129 2) A locally compact field of characteristic p......Page 130 3) Locally compact totally disconnected fields......Page 132 4) Two-dimensional lattices......Page 135 5) The tree of PGL(2,g)......Page 137 References......Page 148 Symbols......Page 154 Index......Page 157 The unitary irreducible representations are classified in three series: a continuous series of spherical, two special representations, and a countable series of cupsidal representations as defined by G.I. Ol'shiankii.