10 WILLIAM H. BARKER

d

TT(X)V = — TT(

exptX)v\

t=0

(2.5)

dt

We call

(TT,V)

the infinitesimal representation of a and K associated to

(TT,V).

Two admissible representations of G, (^1,V'1) and {^VV^ a r e s a ^ t o ^e

infinitesimally equivalent if there is a (a ,K)-module isomorphism between V

x

and V

r

Irreducible unitary representations of G are admissible [13, Theorem 6], and for them

infinitesimal equivalence is the same as unitary equivalence [13, Theorem 8]. Moreover,

(?r,V) is irreducible (V has no proper, closed G-invariant subspaces) if and only if (TT, V)

is irreducible (V has no proper (9 ,K)-invariant subspaces).

If

(TT,V)

is an admissible representation of G, we associate to each f G Ce (G) a

bounded linear operator

00

G

For every f € Ce (G,K) the operator ;r(f) is of finite rank, and we can therefore define a

linear form 0^ on C£°(G,K) by

©„(/) = tr*(/), feC?{G,K).

0^ is called the character of n.

An admissible representation

(TT,V)

is said to be of finite length if there exists a

chain

{0}= ^ c ^ c - - - crB= v

of closed invariant subspaces of V such that the representations on VJ V._x , /=l,2,...,n,

are irreducible. In such a case, the character 0^ extends to an invariant Z -finite distri-

bution on G [21,111.1]. Invariant in this context means

e„(/') - e9V)

for all /€CC°°(G) and X G G, where

f*(v) - f(xyz-\ y£G.

Z-finite means the set {zT | z eZ } spans a finite dimensional space, where Z represents

the collection of bi-invariant differential operators on G (see the following subsection).