4 0. NOTATION AND PRELIMINARIES 2.5

If (TT, V) is a (g, if )-module, then g and K operate as usual on the dual space

V of V. The above conditions are met. The space of if-finite vectors in V' is

then a (g, K)-module, to be called the contragredient (g, if)-module to V, and to be

denoted (7?, V). It is admissible if and only if V is. In that case, V is contragredient

t o F .

A (9, V)-module (71-, V) is unitary if F is endowed with a positive non-degenerate

scalar product ( , ) which is invariant under K and (infinitesimally) invariant under

(ir(k) • v,n(k) • w) — (v,w),

(7r(x)v, W) -h (v,7r(x) • w) = 0 (v,w £V,k £ K,x e g).

We let CQ^K te the category of (9, if )-modules and (9, if )-morphisms, and 11(G)

the set of isomorphisms classes of irreducible admissible (g, K)-modules.

A (g, K)-module (TT, V) (or a differentiable G-module) is said to have an infini-

tesimal character \ if there is a homomorphism Z(g) —* C such that ir(z) = x(z)Td

for all z G Z(g). This is in particular the case if (TT, V) is irreducible and admissible.

2.6. Let (7r, V) G CQ- Then Vo is a (9, if )-module. We denote it sometimes

(TTOJ^O)-

It is admissible (resp. unitary) if

(TT,

V) is so, and it is finitely generated

as a g-module if (TT, V) is finitely generated as a G-module.

It is known that every irreducible admissible (g, if)-module can be realized as

the space of if-finite vectors in an irreducible admissible differentiate G-module

[77]. In fact, this statement is true more generally for finitely generated admissible

(g,if)-modules, but we shall not need this fact.

Two smooth representations are infinitesimally equivalent if the two associated

(g, if )-modules of if-finite vectors are isomorphic.

2.7. We let

Z(Q,K)

denote the subgroup of elements of the center of if which

act trivially on g. If G is connected, with compact center, then Z(g, if) is just the

center of G. We say that a (g, if )-module (71", V) has a central character LO^ if there

exists a character uon : Z(g, if) — » C* such that TT(Z) = ujn(z) -Id for all z G Z(g, if).

If (TT, V) is admissible and irreducible, then it has both an infinitesimal character

and a central character.

2.8. The set of equivalence classes of irreducible unitary representations of G

is denoted £(G) or G.

Let (71", V) be unitary, irreducible. There exists then a unitary character UJ^

of C(G) such that TT(Z) = ujn(z)Id for z G C(G). Therefore \cu,v\ (u,v G V) is a

function on G/C(G). The representation TT is said to be in the discrete series if it is

unitary, irreducible and if its coefficients are square integrable modulo the center,

i.e. on G/C(G). We let £d(G) be the set of equivalence classes of discrete series

representations of G.

If G is compact, then £{G) — £d(G).

3. Linear algebraic and reductive groups

3.0. In this book, up to Chapter XII, we are mainly concerned with real or

complex Lie groups. The point of view of algebraic groups becomes more prominent

in XIII, XIV. Our general reference for linear algebraic groups is [124]. We review

some basic concepts in characteristic 0.