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).
Previous Page Next Page