!AS/Park City Mathematics Series
Volume 12 , 2002Automorphic Forms on Reductive Groups
Armand Borel
Introduction
The goal of these notes is the basic theory of automorphic forms on reductive
groups, up to and including the convergence of Eisenstein series for large values of
the parameters.
The book [6] was written with the general case in mind. Familiarity with it will
be helpful. We shall also refer to it for proofs valid with little or no modification
in the general case.
- Notation
1.1. Let X be a set and f, g strictly positive real functions on X. We write f -< g
if there exists a constant c > 0 such that f(x) :::; cg(x) for all x E X; simila rly,
f >-g if g -< f , and f :::=:: g if f -< g and g -< f.
1.2. Let G be a group. The left (resp. right) translation by g E G is denoted lg
(resp r g); these act on functions via
(1) lg· f(x) = f(g-^1 x), rg · f(x) = f(xg)
If A is a subset of G, then g A = g.A.g-^1 and
NA = {g E G I g A = A}'
ZA = {g E G I ga =a, (a EA)}.1.3. Let G be a Lie group and g its Lie algebra. The latter may be viewed as the
tangent space T 1 (G) at the identity, or as the space of left-invariant vector fields
on G. If X 1 E T 1 ( G), the associated vector field is x f-t x · X 1. The action of X on
functions is given by
(^1) School of mathematics, Institute for Advanced Study, Einstein Dr., Princeton, NJ 08540.
@2007 American Mathematical Society
7