## 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: 21 February 2014

## Notes

Chapter I. The material here is all standard. We have drawn largely on the account in [Tam1960].

Chapter II. For proofs of the assertions in (2.1)-(2.3), see Bourbaki [Bou1968]. The rest of Chapter II is due to Bruhat and Tits: see [BTi1966], [BTi1967]
for summary accounts of their theory.

Chapter III. This chapter is for the most part a transcription into the present context of Satake's paper [Sat1963]. In particular, theorems (3.3.6), (3.3.12)
and (3.3.14) are due to him.

Chapter IV. The formula (4.1.2) for the case of a Chevalley group was announced in [Mac1968-2]. In its present form it dates from 1968 but appears here for the
first time. (4.7.1) is strictly analogous to the real case (see [HJo1969]).

Chapter V. The fact that the support of the Plancherel measure can be bigger than $\stackrel{\u02c6}{T}$ was first remarked by
Matsumoto [Mat1969]. I had neglected this possibility in my original calculations.

## 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.

page history