On the Characters and the Plancherel Formula of Nilpotent Groups ...

270 PUKANSZKY
image, through w, of dbl. Then dA is an invariant measure on A. To
see this, it suffices to observe, that, for any k in K, we have
p(k) 4; = 8; + &, where hi E Hl, and therefore, if g is continuous of
compact support on Y, we obtain
Using dA, (5) can be rewritten as
We recall, that K,(y, y) is continuous and nonnegative on I’. Using (3),
we obtain finally
(d) Observe, that de, = dA dy is an invariant measure on G/S;
we denote by the same symbol the corresponding measure on 0.
Using the notations of Lemma 1 we set dz = dz, dz, *+* dz, . Taking
into account the remarks made before Lemma 1 we see that dz, too,
is an invariant measure on 0, and therefore it differs only by a positive
factor from dv. From this we conclude that, in order to prove Lemma
2, it suffices to show that the function +(z, , za ,..., zd), obtained from
$(e’) by replacing d’ through the parametrization of 0, is summable
with respect to the Lebesque measure on Rd. Since ~(4’) is C” and
has a compact support, +(e’) is rapidly decreasing, and therefore there
exists a constant C, such that
for all z in R”. But by virtue of the properties of the polynomials {Pj},
described in Lemma 1, the right-hand side is certainly summable.
We recall finally, that ~(a) = ($- x 4) (a) (a E G), and therefore
T, = T,*T,.
Remark. Observe, that given a Haar measure da on G, and a
translation-invariant measure d/ on 9, Lemma 2 yields the following