Based on a series of lectures given by Harish-Chandra at the Institute for Advanced Study in 1971-1973, this book provides an introduction to the theory of harmonic analysis on reductive p-adic groups.
Originally published in 1979.
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Introduction to Harmonic Analysis on Reductive P-adic Groups
Based on lectures by Harish-Chandra at The Institute for Advanced Study, 1971-73
By Allan J. SilbergerPrinceton University Press and University of Tokyo Press
Copyright © 1979 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08246-2Contents
Foreword, i,
Acknowledgments., i,
Chapter 0. On the Structure of Reductive p-adic Groups., 1,
Chapter 1. Generalities Concerning Totally Disconnected Groups and Their Representations., 14,
Chapter 2. Jacquet' s Theory, Bruhat' s Theory, the Elementary Theory of the Constant Term., 78,
Chapter 3. Exponents and the Maass-Selberg Relations., 116,
Chapter 4. The Schwartz Spaces., 147,
Chapter 5. The Eisenstein Integral and Applications., 221,
References., 362,
Selected Terminology., 365,
Selected Notations., 370,
CHAPTER 1
On the Structure of Reductive p-adic Groups.
The theory to be developed in the five later chapters of these notes depends upon the structure theory for reductive groups with points in a p-adic field. Fortunately, this theory has been worked out in detail (cf., [2a], [2c] for the essentially algebraic aspects and [4c] for the directly related topological part). In this chapter we very briefly review only those facts from the structure theory which we shall need later. The reader may consult the references both for proofs and more details.
If G is any group and H a subgroup, we write ZG (H)[NG(H)] for the centralizer [normalizer] of H in G. Given x, g [member of] G, we write xg = gxg-1 and Hg = gHg-1.
We write [X] for the cardinality of a finite set X.
For any ring R we write RX for its group of units. If n is a positive integer and R is a commutative ring, we write GLn(R) for the group of all n x n matrices (cij)1 ≤ i, j ≤ n with entries cij [member of] R and determinant in RX. We write det(x) or det(cij) to denote the determinant of a matrix x or (cij).
We write Z, for the ring of ordinary (rational) integers, Q, R, and C for the fields, respectively, of rational, real, and complex numbers.
§0.1. Some Definitions and Facts.
Let Ω be a field and [bar.Ω] an algebraic closure of Ω. For any positive integer m the space [bar.Ω]m of all m-vectors with components in [bar.Ω] carries the Zariski and Ω topologies: A subset S of [bar.Ω]m is called Zariski closed [Ω-closed] if S is the zero set of some finite set of polynomials in [bar.Ω][x1, ..., xm]Ω[x1, ..., xm]]. The complement of a Zariski closed [Ω-closed] subset of [bar.Ω]m is termed Zariski open [Ω-open]. Obviously, the Zariski topology on [bar.Ω]m is finer than the Ω-topology. A subset s1 of a Zariski closed [Ω-closed] set S2 is [subset] [bar.Ω]m called [Zariski dense [[Omega]-dense] in S2 if every Zariski closed [Ω-closed] subset of [bar.Ω]m which contains S1 also contains S2.
Under an obvious identification, we may, for any positive integer n, regard GLn ([bar.Ω]) as the Ω-open subset det(xij) ≠ 0 of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or, more conveniently for what follows, the Ω-closed subset y det(xij) = 1 of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A linear algebraic group or 1.a.g. is a subgroup G [subset] GLn ([bar.Ω]) which is Zariski closed as a subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If G is an 1.a.g., the group law (x, y) -> x-1y for G is given by a polynomial mapping [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
