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

PLANCHEREL FORMULA OF NILPOTENT GROUPS 261
But for this it suffices to substitute iA in place of h’ in the formula of
Harish-Chandra, and to choose for dA
Observe, that this is the dimension of the representation, correspond-
ing to A in A.
To prove, that the orbit O(A) in the right-hand side of (2) is uniquely
determined, since a(O) = 1, it is enough to show, that if 0, and 0,
are two orbits of u and dv, and dv, are nontrivial invariant measures
on them, and if we have
for all C” functions, vanishing outside a given neighborhood of the
neutral element in 9, then 0, = 0, and dv, = dv, . Let L be a
translation-invariant differential operator on 9. Replacing f first by
Lf, then by an approximate identity in the above equation, we conclude,
that
for any polynomial function P(e) on $4. If 0, and 0, are different,
then, since they are compact and without common point, by an
appropriate choice of P, we can arrange that the left-hand side should
be arbitrarily small, while the right-hand side remains close to the
volume of 0, with respect to dv, which, according to assumption,
is nonzero. Hence 0, = 0,; but then dv, and dv, , too, coincide, and
this finishes the proof of Proposition 1.
Summing up once more, Proposition 1 establishes a l-l correspond-
ence between the family of orbits, determined by elements of the form
X + p (A E A), in 9 and the set of all equivalence classes of irreducible
representations of G in the following fashion: The character xA(h)
equals D(h)/?r(h) t imes the Fourier transform of the positive, o-inva-
riant measure, concentrated on the orbit of h + p, in 9, the total
mass of which equals dA = x,,(O). 0 ur next Proposition shows, that
the K-volume (cf. Section 1) and dA are proportional.
PROPOSITION 2. For a $xed h in A, Zet us denote by V(K) the