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

276 PUKANSZKY
which implies at once that f(p(a) p) =f(p). Hence, in particular,
0 is invariant under p. Therefore it is enough to show, that
E4 = dp(a) IP Ep if q = p(a) p on p E 0. But since
for all G in 9 (cf. Section l), it suffices to verify, that vq maps
n$=, A ek and n$=, A ~(a) ek onto the same element in Ad(Tq). We
have o(a) ek - ek E q C q(q) ( j = j, - l), and therefore to obtain
the desired conclusion it suffices to note, that by virtue of the choice
of the system (e3, e, E q(q) - q-,(q) (j = ji , i = 1, 2 ,..., d), and
thus q-I is spanned by R(q) and {e,; 1 < i < k - l}. Since the
vectors {ek) are independent mod R(p) if p E 0, we have in this case
Q(p) # 0. On the other hand, if p lies outside the set
8, = {p; dim O(p) = d},
then clearly Q(p) = 0. H ence, to finish the proof of the statement
made above, it is enough to show that we have Q(p) = 0 for all p
in 0, - 0. With the notations of the previous lemma, let j be the
smallest integer, for which dim .A@) < di . Then evidently j = j,
for some k = 1, 2,..., d and -Y&(q) = q(q) (j = jk), and thus the
system {ei; 1 < i < k} is dependent modulo R(p), implying Q(p) = 0.
Q.E.D.
LEMMA. 4. Let 9 be a non-Abeliun nilpotent Lie algebra,
9n = 3’ 3 ,EpW1 3 **a 3 .ZYO = (0) a Jordan-Holder sequence in 9
(dim 4 = jfor j = 0, 1, 2 ,... n) and {ej’; j = 1, 2 ,..., n} a basis in 3”
such that ej, is in q*, - 9j1. Then there exists a nonconstant p-invariant
polynomial function Q(x) on Y, a positive integer d, d indices
0