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

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260 PUKANSZKY Moreover, the orbit O(h) and the measure dv are uniquely determined kY (2). Proof. We start by observing, that xX4 I 44 I2 = QM W) I 44 F2, and therefore, we have for the left-hand side of (2) where we have put On the other hand, writing dA for the volume of O(A) with respect to dv, we have for any g(e) continuous on 9, putting x = X + p, and therefore the right-hand side of (2) gives where = dn s afV> W) dl 2 G(4) = j, exp [;(~(a) k’, A)] da. Since this function is clearly o-invariant, we can conclude that Summing up, to establish the equality appearing in the Proposition, it suffices to show, that Q,(h) ZE d&h) G(h) (h E b), or that dp(h) j, exp [~(u(cz) h, A)] da = C E(S) exp [i(sh, A)]. SSW
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-invariant 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
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Thus where SIZES OF COMPACT SUBSETS
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SCATTERING THEORY WITH TWO HILBERT
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Condition 10.2. SCATTERING THEORY W
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STRUCTURE SPACES AND EXTENSIONS 0~
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STRUCTURE SPACES AND EXTENSIONS OF
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