Harmonic Analysis and Representation Theory for Groups Acting on Homogenous Trees

✍ Scribed by Alessandro Figá-Talamanca, Claudio Nebbia

• Publisher: Cambridge University Press
• Year: 1991
• Tongue: English
• Leaves: 161
• Series: London Mathematical Society Lecture Note Series
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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