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
It is admissible (resp. unitary) if
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
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.
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