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

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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 dimension 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
PLANCHEREL FORMULA OF NILPOTENT GROUPS 265 be a Jordan-H6lder sequence in 9, and {ej; j = 1, 2,..., n} a basis in Y such that ej’ E S?&., - 9’; (j = 1, 2,..., n). Then there exist n polyno- mials {Pi; j = 1, 2,..., n} of the d variables {xk; k = 1, 2,..., d} and d indices 0
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SIZES OF COMPACT SUBSETS OF HILBERT
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JOURNAL OF FUNCTIONAL ANALYSIS 1, 3
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ALGEBRAS WITH THE SAME MULTIPLICATI
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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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