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

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

Last update: 13 February 2014

## Contents

**Chapter I: Basic properties of spherical functions**
(1.1) The algebra $\mathcal{L}(G,K)$

(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.

Published by the Ramanujan Institute, University of Madras, and printed at the Baptist Mission Press, 41A Acharyya Jagadish Bose Road, Calcutta 17, India.

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