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.
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