Harmonic analysis and representation theory for groups acting on homogeneous trees / Alessandro Figà-Talamanca and Claudio Nebbia.

Format:

Book

Author/Creator:

Figà-Talamanca, Alessandro, 1938- author.

Nebbia, Claudio, author.

Series:

London Mathematical Society lecture note series ; 162.

London Mathematical Society lecture note series ; 162

Language:

English

Subjects (All):

Automorphisms.

Harmonic analysis.

Representations of groups.

Physical Description:

1 online resource (ix, 151 pages) : digital, PDF file(s).

Other Title:

Harmonic Analysis & Representation Theory for Groups Acting on Homogenous Trees

Place of Publication:

Cambridge : Cambridge University Press, 1991.

Language Note:

English

Summary:

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:

Cover; Title; Copyright; Contents; Preface; Chapter I; 1) Graphs and trees; 2) The free group as a tree; 3) Automorphisms of a tree; 4) The group of automorphisms Aut(X); 5) Compact maximal subgroups; 6) Discrete subgroups; 7) Cayley graphs which are trees; 8) Amenable subgroups; 9) Orbits of amenable subgroups; 10) Groups with transitive action on the boundary; 11) Notes and remarks; Chapter II; 1) Eigenfunctions of the Laplace operator; 2) Spherical functions; 3) Intertwining operators; 4) The Gelfand pair (G,K); 5) Spherical representations

6) The resolvent of the Laplace operator and the spherical Plancherel formula7) The restriction problem; 8) Construction and boundedness of Pε; 9) Approximating the projection P0; 10) The constant 1 is a cyclic vector; 11) Notes and remarks; Chapter III; 1) A classification of unitary representations; 2) Special representations; 3) Cuspidal represent at ions and the Plancherel formula of Aut (X); 4) Notes and remarks; Appendix; 1) p-adic fields; 2) A locally compact field of characteristic p; 3) Locally compact totally disconnected fields; 4) Two-dimensional lattices; 5) The tree of PGL(2,g)

ReferencesSymbols; Index

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88477-8

1-107-36671-2

1-107-37139-2

1-107-36180-X

1-107-36821-9

1-299-40446-4

1-107-36425-6

0-511-66232-7