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

358 KATO
with
x* = g (C, + c-) x & + (C, - CJ qy,
y*=r~(c+-c~)B1S+;(C++C-)y.
(9.10)
Here Bily denotes, rather improperly, the element of ID(&) orthogonal
to R(B,) such that B,(Bily) = y (note that y E R(B,) by assumption).
Set
Then by the definition of J
hence
JW> lx, Yl = J@l(Q WI = MG @>I~
&%W = ~&%W~ %@) = W),
II Jut> 1x9 Y> - U&) lx* 9 Yi> II2
= II @z(t), %W> - h&>, ~a&)~ II2
= II M4t> - %&)) ll”R + II 4(t) - 4*(t) IIt3 --+ 0
(9.11)
(9.12)
(9.13)
as t -+ & co, for the second term in the last member tends to zero
by (9.9) and the same is true of the first term because by (9.12) it is
equal to
note (9.9) and that B,u,,(t) E [%(B,)].
Since the {x, JJ> with the above properties (x E D(B,) n QIR,
Y E WV
continuity
n Ql@ f orm a dense subset of the subspace of absolute
for Hr (see Lemma 8.1), (9.13) implies the existence of
IV, = W,( U, , Ul; J), with
W&, y> = {a 3 Y*)
(9.14)
for the {x, JJ} restricted as above. (9.10) does not make sense for a
general {x, r} E !& . But it can be extended to all {x, y) E I’#, (which
is equivalent to X, y E QIR by Lemma 8.1) if Bil and B, on the right
are replaced by &l and 8, , respectively, for the ensuing map from
P#, to P&, is bounded. Furthermore, (9.14) is then even true for
every (x, r} E& in virtue of the property C, = C&r . In what
follows (9.10) should be read in the extended sense so that (9.14) is
true for all {X, r} E .Eil .