2 0. NOTATION AND PRELIMINARIES 1.5

1.5. The Lie algebra of a real Lie group G,H,-— will be denoted by the

corresponding German lower case letter g1 f), • • •, and the exponential map g — G

is denoted exp. We also write ex for expx(x eg). If m is a subspace of g, then mc

stands for the complexification m ®R C of m.

The universal enveloping algebra over C of g is denoted U(g). Its center is

denoted Z(g).

The centralizer (resp. normalizer) of m in g is denoted ^(m) or 30(m), resp. n(m)

orn0(m):

3s (m) = {x e g/[m,x] = 0}, ng(m) = {x e g/[x,m] em (me 2tt)}.

As usual the differential of Intx (x e G) at 1 is denoted Adx, and, for x e g,

adx: g H^ g is defined by adx(y) = [x,y\. For m C g, we let

^c(tn) = {x e G | Adx(m) = m (m e m)},

J\fG(m) = {x e G | Adx(m) = m}.

1.6. If G is a Lie group, then X(G) is the group of continuous homomorphisms

of G into R* and

°G= p| ker|X|.

xex(G)

It is a normal subgroup which contains the derived group and all compact

subgroups of G.

1.7. Unless otherwise stated, topological vector spaces are assumed to be over

R or C, Hausdorff locally convex and quasi-complete, and manifolds to be C°° and

countable at infinity. If M is a manifold and V a topological vector space, then

C°°(M; V) is the space of C°°-funetions of M, with values in V, endowed with the

C°°-topology. The space of ^/-valued smooth differential p-forms (p e N) on M is

denoted

AP(M;

V), and A*(M; V) is the direct sum of the spaces

AP(M;

V). Thus

A°(M; V) = C°°{M; V). If V is a Frechet space, then so is AP(M- V) (pGN).

If M, A^ are manifolds, then C°°(A,B) is the space of smooth maps A — B,

endowed with the C°°-topology.

2. Representations of Lie groups

2.1. Let G be a Lie group with finite component group. By a topological

G-module (or simply a G-module) V, where V is assumed to be a locally convex

and locally complete Hausdorff topological vector space over C, we mean a ho-

momorphism G — Aut V defined by a continuous map G x V —» V. It will be

denoted (TT, V), or V or TT. The action of g on v is often denoted g.v or gv rather

than n(g)v. We shall denote by CQ the category of topological G-modules and

equivariant continuous linear maps.

V is said to be finitely generated if there is a finite subset S of V such that the

span of the vectors g.c (g e G,c e S) is dense in V.

2.2. Let

(TT,V)

G

CG-

For v e V we let cv: G — V denote the orbit map

cv(g)

— K(g)v. It is continuous. If v is a continuous functional on V, then the

function cv$ on G defined by

(1) CvAd) = (A9)v,v) = (cv(g),v) (g e G)

is called a coefficient of TT.