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

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 $\stackrel{ˆ}{T}$ 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.