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

264 PUKANSZKY
2 are chosen in such a fashion, that the ratio da/d/ at the neutral
element of 2 should be the same as V(K)/d, = C, 2” * m! nasP+ (a, p).
We shall see in the next section that, in the nilpotent case the value
of the analogously defined constant is 4” * m! rrm.
3. In this Section we consider again a nilpotent Lie algebra P’ of
dimension n > 0, with the corresponding connected and simply-
connected group G. We denote by u the adjoint representation of G,
and by p the representation, which is contragredient to u, on 2’.
In other words, denoting by (/, el) the canonical bilinear form on
2 x ZZ”, we have, for all a in G,
(u(a) rt, P) = (l, p(a-‘) 8’) (t E Y, t’ E 9’).
Let H be a subalgebra of 2, of a dimension h < n. We shall say,
that the ordered (n - h)-tuple of elements {t; , /a ,..., &‘n-h} of $P is a
supplementary basis of H, provided for each j = 1, 2,..., n - h the
subspace, spanned by Hand (8, ,..., e,>, of 2 is a subalgebra of dimen-
sion h + j. Let us denote by exp H the subgroup, which is the image
of H through the exponential map, of G and let us set gj(t) = exp (J’+)
(j = 1, 2,..., 12 - h). The map of exp H x Rn-h into G, which
assigns to (h, (ti , t, ,. .., tn-h)) (h E exp H) the element
of G, is a homeomorphism between exp H x Rn-h and G ([4J p. 96,
Remarque 2), and dt, dt, *a+ dt,-, defines an invariant measure on the
homogeneous space of right-classes of G according to exp H ([4J,
p. 121, Remarque).
An element c!; of 2’ having been fixed, we shall denote by B(lr ,&J
the skew-symmetric bilinear form on 9 x 2 defined by ([/r , t.J, 8;)
(/i , tz E 2). Given a subspace H of JZ’, we shall write (H)i for its
orthogonal complement, with respect to B, in 2, and H-L for its
orthogonal complement, with respect to the canonical bilinear form
in 9’.
We start with a new proof for a statement, announced first by
Kirillov ([4], Theo&me on p. 50, and also the Remark below).
LEMMA 1. Let & be a nonzero element in 9”, and let 0 be its orbit
with respect to p; we set d = dim 0, and suppose d > 0. Let