8 WILLIAM H. BARKER
ne = (J *) = exp(ey), £eR.
Particular subgroups of G are defined by
K = {*, : 0GR}, A = {a, : *ER}, iV = {n^ : $€R}.
The corresponding subalgebras of a will be denoted by k , a , and rt respectively. K is a
maximal compact subgroup of G. The Iwasawa decomposition for G gives that the map-
ping (k,a,n) -• kan is a difFeomorphism of K x A x N onto G. If x = kan, we will use
the notation
k(x)=k, h(x) = a.
Let A+ = {af : t0} and A~ = {at : t0}. The Cartan decomposition for G gives
G = K'Cl{A+)*K
i.e., each x £ G can be decomposed into the form x = k&atk,. The at term is unique,
which allows us to write
H(x) = t. (2.2)
Let P be the subgroup of SL(2, R) consisting of matrices of the form
a b
0
a"1
, a€R , 6GR,
and let M = I}, where I is the identity matrix. Then
P=MAN.
For a e G let La denote the left translation map x-+ax and Ra the right translation
map x-xa . We further define the character a : A - R by
a(at) = e , t€R.
The groups K, A, N and G have biinvariant Haar measures which we normalize as
follows:
dk = dk$ = d0/2w, 0 ^ 0 ^ 2?r,
da = dat = d*, i€R,
dn = cfn^ = d£, £eR,
dz = a(a)dfc (fa dn.
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