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
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) =
Intg is the inner automorphism x i- g x
We also write
for Int #(x),
- 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 is the connected component of the
identity in G.
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