CHAPTER 0

Notation and Preliminaries

§1 contains some general notation, §2 some definitions and facts on representa-

tions of Lie groups, and §3 fixes a number of conventions on reductive groups. The

notation introduced here will often be used without reference.

1. Notation

1.1. As usual, Z is the ring of integers, N = {z G Z | z 0} the set of

natural integers, Q (resp. R, resp. C) the field of rational (resp. real, resp. complex)

numbers, R+ the multiplicative group of strictly positive real numbers.

If A is an algebra with identity, then A* is the group of units of A.

1.1.1. If V = 0

i e Z

V1

is a vector space graded by Z and if m G Z, then V[m]

denotes the graded vector space defined by

V[mY = Vl+rn (i G Z).

1.1.2. Let V be a complex vector space. If V has the structure of a module

over a group or a Lie algebra and if m G N, then we have consistently written mV

for the direct sum of m copies of V, with the corresponding diagonal action, thus

committing an abuse of notation. To adopt a correct one would entail an amount

of changes that we found too daunting. We thereby, regretfully, announce that we

shall maintain our original convention.

1.2. If G is a group, and M a subset of G, then ZG{M) or Z(M) is the

centralizer of M and NG{M) or M(M) the normalizer of M:

ZG(M) = {geG\g'm = m.g(me M)},

ArG(M) =

{geG\9'M.g-1cM}.

Intg is the inner automorphism x i- g • x •

g~l.

We also write

9x

for Int #(x),

and

gM

- Intp(M). The center of G is denoted Z(G) or C(G), and VG is the

derived group of G.

1.3. If g G G, then £g (resp. rg) denotes the left (resp. right) translation by

g on G, or on functions / on G. In particular

(1) £gf(x) = fig-1 • *), rgf(x) = f(x . g) {x G G).

Thus ig.h - ig • 4 , rgh =rg-rh (g, h G G).

1.4. If G is a topological group, then G° is the connected component of the

identity in G.

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http://dx.doi.org/10.1090/surv/067/01