## Spherical Functions on a Group of $p\text{-adic}$ Type

Last update: 13 February 2014

## Contents

Chapter I: Basic properties of spherical functions (1.1) The algebra $ℒ\left(G,K\right)$
(1.2) Spherical measures and spherical functions
(1.3) Induction of spherical functions
(1.4) Positive definite spherical functions and representations of class 1
(1.5) Fourier analysis

Chapter II: Groups of $p\text{-adic}$ type (2.1) Root systems
(2.2) Affine roots
(2.3) BN-pairs
(2.4) Buildings
(2.5) Groups with affine root structure
(2.6) Cartan and Iwasawa decompositions
(2.7) Groups of $p\text{-adic}$ type

Chapter III: Spherical functions on a group of $p\text{-adic}$ type (3.1) The root system ${\Sigma }_{1}$
(3.2) Haar measures
(3.3) Zonal spherical functions on $G$ relative to $K$

Chapter IV: Calculation of the spherical functions (4.1) Statement of result
(4.2) Prehminary reductions
(4.3) The integral ${\int }_{U\left(a\right)}{\varphi }_{s}\left({u}_{a}\right)d{u}_{a}$
(4.4) Reduction of ${J}_{w}\left(s\right)$
(4.5) End of the proof
(4.6) The singular case
(4.7) Bounded spherical functions

Chapter V: Plancherel measure (5.1) The standard case
(5.2) The exceptional case
(5.3) Comparison with the real and complex cases

## Notes and references

This is a typed version of the book Spherical Functions on a Group of $p\text{-adic}$ Type by I. G. Macdonald, Magdalen College, University of Oxford. This book is copyright the University of Madras, Madras 5, India and was first published November 1971.