Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Mathematics EXPLICIT ORBITAL PARAMETERS AND THE PLANCHEREL MEASURE FOR EXPONENTIAL LIE GROUPS BRADLEY N. CURREY Volume 219 No. 1 March 2005

2 PACIFIC JOURNAL OF MATHEMATICS Vol. 219, No. 1, 2005 EXPLICIT ORBITAL PARAMETERS AND THE PLANCHEREL MEASURE FOR EXPONENTIAL LIE GROUPS BRADLEY N. CURREY Lebesgue measure on the linear dual of the Lie algebra of an exponential solvable Lie group is decomposed into semi-invariant orbital measures by means of a detailed analysis of orbital parameters and a natural measure on an explicit cross-section for generic coadjoint orbits. This decomposition yields a precise and explicit description of the Plancherel measure. Introduction For an exponential solvable Lie group G, the classical Plancherel formula for nonunimodular groups [Duflo and Moore 1976] is combined with the method of coadjoint orbits to construct an orbital Plancherel formula [Duflo and Raïs 1976]. Given a choice of a semi-invariant positive Borel function ψ on the linear dual g of the Lie algebra g, measurable fields {π, } g /G of irreducible representations and {A ψ, } g /G of positive self-adjoint, semi-invariant operators (transforming by the square root of the modular function) in, and the Borel measure m ψ on g /G are constructed so that for the usual class of functions φ on G, (0.1) φ(e) = g /G Tr ( A 1 ψ, π(φ) A 1 ψ, ) dmψ ( ). holds. Though each of the measurable fields above depends upon the choice of ψ, the object {A 2 ψ, dm ψ( )}, which is interpreted as a measure on positive, semiinvariant operator fields over Ĝ = g /G, is canonical, and is referred to as the Plancherel measure. In the nilpotent case, where one takes ψ 1 and A ψ, Id, the measure m ψ = m is described precisely by L. Pukánszky [1967]. Let {Z 1, Z 2,..., Z n } be a basis of g where for each 1 j n, the -span of Z 1,..., Z j is an ideal in g. Let g have Lebesgue measure d X obtained by its identification with n via this basis, let g have the Lebesgue measure via its dual basis, and let G have the Haar measure d(exp X) = d X. Given these initial choices, Pukánzsky gives an algorithm for MSC2000: 22E25, 22E27. Keywords: coadjoint orbit, representation, polarization. 97

3 98 BRADLEY N. CURREY computing the Plancherel measure. For each l g, define the jump index set e(l) by e(l) = {1 j n g j g j 1 + g(l)} where g(l) is the stabilizer subalgebra for l. One has e(l) = dim( l ); among those l whose orbits have maximal dimension 2d, where d is a nonnegative integer, view the sets e(l) as increasing sequences and order them lexicographically. Let e = {e 1 < e 2 < < e 2d } be the minimal jump index sequence, and = {l g e(l) = e}. The set is G-invariant and Zariski open in g. Also associated with e are the skew-symmetric matrices M e (l) = [ l([z ea, Z eb ]) ] 1 a,b 2d, l g, and the subspace V = {l g l(z ea ) = 0 for 1 a 2d}. One has = {l g det(m e (l)) = 0} and = V is a topological cross-section for /G. In fact (see [Pukánszky 1967, Lemma 4]) there is an explicit rational map P : 2d such that P(z, sl) = P(z, l) for each z 2d and s G, and that, for each l, P(, l) is a polynomial bijection between 2d and the coadjoint orbit of l. The cross-section = P(0, ) and the restriction of P to 2d is a rational bijection whose Jacobian is one. The basis of the Pukánszky algorithm for the Plancherel formula is the elementary decomposition of Lebesgue measure on g [Pukánszky 1967, p. 279]: (0.2) h(l) dl = h(p(z, λ)) dz dλ, g 2d where dλ is Lebesgue measure on V (when V is identified with n 2d via the dual basis {Z j j / e}), and h is a positive Borel function on g. The inner integral in (0.2) is actually an integral over the coadjoint orbit λ of λ which is G-invariant, and hence is a multiple of the canonical measure β λ on λ. Precise computation of the Plancherel measure is simply a matter of computing this multiple r(λ) for each λ and then plugging that into (0.2). The result is that r(λ) = (2π) d P e (λ) where P e (λ) is the Pfaffian of M e (λ). Equivalently, the measure dm on g /G is given on by (2π) d P e (λ) dλ, and the formula 1 (0.3) φ(e) = (2π) n+d Tr(π λ (φ)) P e (λ) dλ, a simple version of (0.1), is obtained by combining the above with the Kirillov character formula and ordinary Fourier inversion. All this depends of course upon the choice of Jordan Hölder basis made at the outset, but only upon this choice. Independent of this choice one sees that the Plancherel measure, as a measure on the orbit space, belongs to the family of rational measures on g /G.

4 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 99 Suppose now that G is exponential solvable. It is perhaps not surprising that the methods of Pukánszky can be extended to obtain a cross-section for generic coadjoint orbits. However, the execution of this method, and the orbit picture that emerges from it, are more complex. The jump sets e(l) are defined as before only now the basis {Z j } is a basis of the complexified Lie algebra s = g c, for which span{z 1, Z 2,..., Z j } = s j is an ideal in s, and if s j = s j then s j+1 = s j+1 and Z j+1 = Z j. As shown in [Currey 1992; Currey and Penney 1989], the notion of generic orbits must be refined in order to complete the construction of an explicit topological cross-section for the generic orbits. Among other things this involves selecting an index subset ϕ of e, which, roughly speaking, identifies directions in g in which G acts exponentially. Nevertheless, there is an explicit, G-invariant Zariski open subset e, and for l, a precise generalization of the Pukánszky map P(z, l) described above. One still has P(z, sl) = P(z, l) for s G, but now some of the variables z 1, z 2,..., z 2d may be complex variables, and P is not necessarily rational but real analytic. Simultaneously there is an orbital cross-section obtained by fixing the variables z a in an appropriate way. Despite the highly nonalgebraic nature of the coadjoint action here, it is shown that the cross-section is in fact a real algebraic submanifold of g. For each l, there is a real analytic submanifold T (l) of m (depending only on the orbit of l) such that P(, l) is an analytic bijection between T (l) and the coadjoint orbit of l. The result is that has in a very explicit way the structure of a bundle over its orbital cross-section: P T (λ) λ = P, λ λ where P (l) = P(z (l), l) for a particular (G-invariant) choice z (l) T (l). The fiber of the bundle is a cone W 2d that is naturally homeomorphic with each T (l), and local trivializations are given over Zariski-open subsets E of. Given that these constructions are a natural generalization of the Pukánszky parametrization, the question now becomes: what is the appropriate generalization of (0.2) in the exponential case? There are at least two obvious complications: (1) The description of given by the Pukánszky map is not as a simple product, but rather as a bundle over the cross-section ; and (2) is not necessarily (a Zariski open subset of) a subspace V. In [Currey 1992] it is shown that is a smooth, real algebraic submanifold of g, determined by explicit polynomials. Letting S t, for each 1 t n 2d, stand for any of,, 0 = { 1, +1}, or 1, in this paper we show that there is a product S = S 1 S 2 S n 2d

5 100 BRADLEY N. CURREY such that each Zariski-open subset E of over which can be trivialized is naturally identified with a dense open subset of S. These identifications differ with the sets E, but only slightly; in particular, if A is a Borel subset of two trivializing subsets E 1 and E 2, then A is identified via E 1 and E 2 with sets in S of equal Lebesgue measure. Thus carries a natural Lebesgue measure, which we denote by dλ. We then use the bundle structure of to decompose Lebesgue measure dl on g. We show that for each λ there is a semi-invariant measure ω λ on the coadjoint orbit λ through λ, with multiplier, such that (0.4) h(l) dl = h(l) dω λ (l) dλ g for any positive Borel function h. If ψ is any positive semi-invariant function on g with multiplier 1, then dω λ is given by λ dω λ = r ψ (λ) ψ 1 dβ λ, where β λ is the canonical measure on λ, and where r ψ (λ) is defined by r ψ (λ) = P e (λ) ψ(λ) (2π) d j ϕ 1 + iα j. Here ϕ is the index subset of e referred to above (which is empty in the nilpotent case), and 1 + iα j = γ j /R(γ j ), where γ j is the j-th root of the coadjoint action. Just as in the nilpotent case, this yields a description of the Plancherel measure in precise terms. Take (π λ, λ ) to be the irreducible representation induced from the Vergne polarization at λ (corresponding to the Jordan Hölder sequence already chosen). Since the Vergne polarization is contained in the kernel of, the operator D λ defined by D λ f (a) = (a) f (a) for f λ defines a positive self-adjoint semi-invariant operator of weight 1. Using this and the character formula for exponential solvable Lie groups, one has where {A 2 ψ, dm ψ( )} g /G = {K λ dλ} λ, K λ = P e (λ) (2π) n+d j ϕ 1 + iα j D λ. The Pukánszky version of the Plancherel formula becomes φ(e) = Tr ( K 1/2 λ π λ (φ) K 1/2 ) λ dλ 1 = (2π) n+d j ϕ 1 + iα Tr ( D 1/2 λ π λ (φ) D 1/2 ) λ Pe (λ) dλ. j

6 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 101 In Section 1 of this paper we review the relevant results of [Currey 1992], and then proceed with an expansion of these results to obtain more detailed information about the bundle structure in general, and the cross-section in the generic case. In Section 2 this information is used to define Lebesgue measure on and then to deduce the decomposition (0.4) and the description of the Plancherel measure. 1. The Collective Orbit Structure 1.1. Preliminaries. Let g be a solvable Lie algebra over with s = g c its complexification, and choose a basis {Z 1, Z 2,..., Z n } for s with the following properties. (i) For each 1 j n, the space s j = -span{z 1, Z 2,..., Z j } is an ideal in s. (ii) If s j = s j then s j+1 = s j+1 and Z j+1 = Z j. Moreover, in this case, there is A g such that [A, Z j ] = (1 + iα)z j mod s j 1, where α is a nonzero real number. (iii) If s j = s j and s j 1 = s j 1, then Z j g. As in [Currey 1992], it will be convenient to make the following notation: I = {1 i n s j = s j }, and for each 1 j n set j = max ({0, 1,..., j 1} I ) and j = min ({ j, j + 1,..., n} I ). Thus for each j, s j =s j 1 s j 1 and s j =s j +s j. For Z s, denote the real part of Z by RZ, and the imaginary part of Z by IZ. (We also use these symbols to denote real and imaginary parts of a complex number.) Define a basis for g as follows: let X j = Z j if Z j g, and if s j = s j then set X j = RZ j and X j+1 = IZ j. Using the ordered basis {X j } to identify g with n, let d X denote Lebesgue measure on g. Let dl be Lebesgue measure on g obtained via the ordered dual basis {X j }. We regard g as a real subspace of the complex vector space s, and for convenience we denote l(z) = l, Z by lz, for Z s and l g. We identify an element l g with the n-tuple (l 1, l 2,..., l n ), where l j = lz j. For each l g let s(l) = {Z s l[z, W ] = 0, for all Z s}, and let p(l) be the complex Vergne polarization associated with the sequence {s j } chosen. For any l g and any subset t of s, we use the usual notation t l = {Z s l[z, X] = 0 for all X t}. Let G be the unique connected, simply connected Lie group with Lie algebra g; we assume in this paper that G is exponential, meaning that the exponential map exp : g G is a bijection. Let da be the left Haar measure on G defined by d(exp X) = j G (X)d X, where j G (X) = det(1 e ad X )/ad X. Let be the modular function: d(ab) = (b) da. The coadjoint action of G on g extends to an action of G on s and restricts to an action of G on each ideal s j. We denote each

7 102 BRADLEY N. CURREY such action multiplicatively. For each 1 j n, set s j = { l g l(s j ) = {0} }, let µ j : G be defined by s Z j = µ j(s)z j mod s j, and let γ j : g be the differential of µ j. Since G is exponential, there is a real number α j such that γ j = R(γ j )(1 + iα j ), for 1 j n. The results stated in [Currey 1992, Proposition 2.6, Theorem 2.8] provide us with a stratification of the linear dual g of g into Ad (G)-invariant layers and in each layer an explicit description of the space of coadjoint orbits. We summarize the stratification procedure as follows. (1) To each l g there is associated an index set e(l) {1, 2,..., n} defined by e(l) = {1 j n s j s j 1 + s(l)}. For a subset e of {1, 2,..., n}, the set e = {l g e(l) = e} is algebraic and G-invariant, and we refer to the collection of nonempty e as the coarse stratification of g. The coarse stratification has had various applications; see for example [Pedersen 1984]. There is an ordering on the coarse stratification for which the minimal element is Zariski open in g and consists of orbits having maximal dimension. (2) To each l there is associated a polarizing sequence of subalgebras s = h 0 (l) h 1 (l) h d (l) = p(l), and an index sequence pair (i(l), j(l)) having values i(l)={i 1 <i 2 < <i d } and j(l) = { j 1, j 2,..., j d } in e(l), defined for 1 k d by the recursive equations i k = min { 1 j n s j h k 1 (l) h k 1 (l) l}, h k (l) = ( h k 1 (l) s ik ) l hk 1 (l), j k = min { 1 j n s j h k 1 (l) h k (l) }. Then i k < j k for each k, and e(l) is the disjoint union of the values of i(l) and j(l). Note that since i(l) must be increasing, it is determined by e(l) and j(l). For any splitting of e into such a sequence pair (i, j) we set e, j = {l e j(l) = j}. These sets are also algebraic and G-invariant, and we refer to the collection of nonempty e, j as the fine stratification of g. There is an ordering on the fine stratification for which the minimal layer is a Zariski open subset of the minimal coarse layer. (3) Now fix a layer e, j in the fine stratification. For each l e, j, set ϕ(l) = { j e s l j ker(γ j) = s l j ker(γ j) }.

8 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 103 The index set ϕ(l) identifies those directions j in e where the coadjoint action of G dilates by its character µ j. If j ϕ, then j 1 I, and j I if and only if µ j is real. It is easily seen that ϕ(l) is contained in the values of i, and there are examples where ϕ(l) is not constant on the fine layer. For each j i, there is a rational function q j : g such that q j is relatively invariant with multiplier µ 1 j, and such that for l e, j, one has j ϕ if and only if q j (l) = 0. So for each subset ϕ of the values of i, the set e, j,ϕ = {l e, j ϕ(l) = ϕ} is an algebraic subset of e, j. We refer to this further refinement of the fine stratification as the ultrafine stratification of g. The ultrafine stratification also has an ordering for which the minimal layer is a Zariski open subset of the minimal fine layer. (4) Now fix an ultrafine layer = e, j,ϕ and let ι={ j e ϕ j / I and j+1 / e}. Let V 0 be the span of those Z j for which either j / e or j ϕ ι. Then for each i ι, there is a rational function p i : g such that the set = { l V 0 p i (l) = 0 for every i ι, and q j (l) = 1 for every j ϕ } is a topological cross-section for the orbits in Parametrizing an orbit. Take l g and write e(l) = {e 1 < e 2 < < e 2d }. Then, for each j e, one can select X j g (s j s j ) so that (t 1, t 2,..., t 2d ) exp(t 1 X e1 ) exp(t 2 X e2 ) exp(t 2d X e2d )l is an analytic diffeomorphism Q(t, l, X e1, X e1,..., X e2d ) of 2d with the coadjoint orbit of l. The starting point for the constructions of [Currey 1992] is a procedure for selecting the X j, in terms of the elements l belonging to a fine layer e, j, so that the resulting map Q(t, l) is analytic in l and has a manageable and somewhat explicit form. The relevant result is [Currey 1992, Lemma 1.3]; the following lemma is a restatement of the important aspects of this result in a somewhat simplified form. We then include a description of the procedure by which this result is proved in [Currey 1992]. Finally, we show how this result is used to define the orbit parametrization, and we observe that a slight modification of the selection procedure in [Currey 1992] obtains a parametrization that is simpler in some cases. Lemma [Currey 1992, Lemma 1.3]. Let g be an exponential solvable Lie algebra over, and choose a good basis for s = g c. Let e, j be a fine layer. Then there is a cover F = {O t } of e, j by finitely many Zariski open sets, and for each O F and 1 k d, there are analytic functions X k : O g, Y k : O g, and φ k : O 1 with the following properties. (i) l[x j (l), X k (l)] = l[y j (l), Y k (l)] = 0 for 1 j, k d. (ii) l[x j (l), Y k (l)] = 0 if and only if j = k, for 1 j, k d.

9 104 BRADLEY N. CURREY (iii) For each k, the functions l φ k (l)x k (l) and l φ k (l)y k (l) extend to rational functions from e, j into s and are independent of O. (iv) For each 1 k d, set m k (l) = -span { φ 1 (l)y 1 (l), φ 2 (l)y 2 (l),..., φ k (l)y k (l), φ 1 (l)x 1 (l), φ 2 (l)x 2 (l),..., φ k (l)x k (l) }, so that s = m k (l) m k (l) l for each l. For Z s and l, let ρ k (Z, l) be the projection of Z into m k (l) l parallel to m k (l), with ρ 0 (Z, l) = Z. Then X k (l) and Y k (l) are in the image of ρ k 1 (, l), and the function ρ k is defined recursively by the formula (1.2.1) ρ k (Z, l) = ρ k 1 (Z, l) l[ρ k 1(Z, l), X k (l)] l[y k (l), X k (l)] Y k (l) l[ρ k 1(Z, l), Y k (l)] l[x k (l), Y k (l)] X k (l). (v) For each l, ρ k (s j, l) s j for 1 j n and 0 k d, and X k (l) s j Y k (l) s i. k (vi) For 1 k d, X k (l) has the form X k (l) = R ( l[ρ k 1 (Z jk, l), Y k (l)]ρ k 1 (Z jk, l) ). Remark In the construction of [Currey 1992, Lemma 1.3], one actually has X k (l) = a(l) R ( l[ρ k 1 (Z jk, l), Y k (l)]ρ k 1 (Z jk, l) ), where a(l) is a real-valued analytic function on O. Formula (vi) above represents a simplification of the procedure there. For the purposes of this paper it will be necessary to analyze the preceding objects in some detail, so we recall how these objects are defined. Let 1 k d. If k > 1, assume that a Zariski open subset O of e, j has been selected, and that Y 1, Y 2,... Y k 1, X 1, X 2,..., X k 1 have been defined so as to satisfy (i) (vi) above, so that we have the map ρ k 1. If k = 1, set O = e, j and ρ 0 (Z, l) = Z for Z s and l g. We then proceed to select a Zariski open subset of O and to construct Y k and X k. We consider several cases. In each of them X k (l) is defined essentially as in Lemma 1.2.1(vi) above, although in Cases 3 and 5, Remark applies. In those cases we justify the remark. Case 0. i k I and i k 1 I. Here Z ik g, and we set Y k (l) = ρ k 1 (Z ik, l). The rest of the cases are those for which Z ik = Z ik. k,

10 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 105 Case 1. i k / I and i k + 1 / e. Here one finds that the complex numbers β 1,k (l) = l[ρ k 1 (Z jk, l), RZ ik ] and β 2,k (l) = l[ρ k 1 (Z jk, l), IZ ik ], satisfy I ( β 1,k (l)β 2,k (l) ) = 0. Write O = O 1 O 2, where O t = {l O β t,k (l) = 0}. For l O t, set φ t,k (l) = β t,k(l) β t,k (l), and Y t,k (l) = φ t,k (l) 1( β 1,k (l)ρ k 1 (RZ ik, l) + β 2,k (l)ρ k 1 (IZ ik, l) ), t = 1, 2. Case 2. i k 1 = j r / I. Here we set Y k (l) = ρ k 1 ( X r (l), l) where X r (l) = I ( l[ρ r 1 (Z jr, l), Y r (l)]ρ r 1 (Z jr, l) ). Case 3. i k / I and i k + 1 = j k. Here Y k (l) = ρ k 1 (IZ ik, l) and in the proof of [Currey 1992, Lemma 1.3], X k (l) = ρ k 1 (RZ ik, l). Note that R ( l[ρ k 1 (Z jk, l), Y k (l)]ρ k 1 (Z jk, l) ) so that Remark holds. = l[ρ k 1 (RZ ik, l), ρ k 1 (IZ ik, l)] ρ k 1 (RZ ik, l) Case 4. i k / I, i k + 1 = i k+1. This case splits into two subcases. Case 4a. Z jk+1 = Z jk. Here Y k (l) = ρ k 1 (RZ ik, l). Case 4b. Z jk+1 = Z jk. This case is just like Case 1: the functions β t,1 (l), and the sets O t, t = 1, 2 are defined exactly the same way, as is Y t,k (l), for l O t, t = 1, 2. Case 5. i k 1 = i k 1 / I. Again there are two subcases. Case 5a. Z jk 1 = Z jk. Set r = k 1 and note that Case 4 holds for r. We have Y k (l) = ρ r (IZ ir, l), and in the proof of [Currey 1992, Lemma 1.3], X k (l) is defined as X k (l) = ρ k 1 ( X r (l), l) where X r (l) = I ( l[ρ r 1 (Z jr, l), Y r (l)]ρ r 1 (Z jr, l) ). We claim that Remark holds in this case also. Set β r (l) = l[ρ r 1 (Z jr, l), Y r (l)]; then ρ r 1 (Z jr, l) = β r(l) ( Xr β r (l) 2 (l) + i X r (l) )

11 106 BRADLEY N. CURREY and Now ρ r 1 (Z jk, l) = ρ r 1 (Z jr, l) = β r(l) β r (l) 2 ( Xr (l) i X r (l) ). ρ r (Z jk, l) = ρ r 1 (Z jk, l) l[ρ r 1(Z jk, l), X r (l)] Y r (l) l[y r (l), X r (l)] l[ρ r 1(Z jk, l), Y r (l)] X r (l) l[x r (l), Y r (l)] = β r(l) β r (l) 2 ( Xr (l) i X r (l) ) iβ r(l) β r (l) 2 l[ X r (l), X r (l)] l[y r (l), X r (l)] Y r(l) = iβ r(l) β r (l) 2 ( X r (l) l[ ) X r (l), X r (l)] l[y r (l), X r (l)] Y r(l), and so because l[y k (l), Y r (l)] = 0, we get β r(l) β r (l) 2 X r(l) R ( l[ρ r (Z jk, l), Y k (l)]ρ r (Z jk, l) ) ( ) iβr (l) = R β r (l) 2 l[ X r (l), Y k (l)] ρ r (Z jk, l) ( ( ( iβr (l) iβr (l) = l[ X r (l), Y k (l)] R X β r (l) 2 β r (l) 2 r (l) l[ ))) X r (l), X r (l)] l[y r (l), X r (l)] Y r(l) = l[ ( X r (l), Y k (l)] X β r (l) 2 r (l) l[ ) X r (l), X r (l)] l[y r (l), X r (l)] Y r(l) = l[ X r (l), Y k (l)] β r (l) 2 ρ r ( X r (l), l). This proves the claim. Case 5b. Z jk 1 = Z jk. Again we set r = k 1. Then Case 4b holds for r and we have Y k (l) = β 1,r (l)ρ r 1 (IZ ir, l) β 2,r (l)ρ r 1 (RZ ir, l). (In this subcase X k (l) is defined in [Currey 1992, Lemma 1.3] exactly as Lemma 1.2.1(vi) above.) Write e = {e 1 < e 2 < < e 2d } and fix O F. In [Currey 1992, Proposition 1.5], the objects X k (l) and Y k (l) are used to define analytic functions r a : O g

12 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 107 for the purpose of parametrizing the orbit of each l O in a manageable way. The definition given there is X k (l) l[zea, X k (l)] r a (l) = Y k (l) l[z ea, Y k (l)] if e a = i k, if e a = j k. Suppose that j = e a e with j 1 I. If also j I, then ad r a (l)l = ζ a (l)z j mod s j, where ζ a (l) = ±1 (and is constant on O). If j / I, then ad r a (l)l = ζ a (l)z j + ζ a(l)z j+1 mod s j+1, where ζ a (l) is a complex number of modulus one. (Recall Z j+1 = Z j in this case.) If also j + 1 = e a+1 e, then similarly ad r a+1 (l)l = ζ a+1 (l)z j + ζ a+1(l)z j+1 mod s j+1 Note that ζ a (l) = l[z j, r a (l)] (and ζ a+1 (l) = l[z j, r a+1 (l)] if j + 1 e), so that l ζ a (l) (and l ζ a+1 (l)) are analytic functions on O. It is shown in [Currey 1992, Proposition 1.5] that if j / I and both j = e a and j + 1 belong to e, then for each l the complex numbers ζ a (l) and ζ a+1 (l) are linearly independent over. It will simplify a subsequent computation if we can show that in fact they are orthogonal, that is, that R ( ζ a (l)ζ a+1 (l) ) = 0. To do this it is necessary to alter (slightly) the definition of r a (l) in one particular case: suppose that e a = i k and that Case 4a holds for k. In other words, suppose that e a = i k / I, that i k + 1 = i k+1, and that Z jk+1 = Z jk. Then I claim that we could have defined the X k (l), X k+1 (l), Y k (l) and Y k+1 (l) as follows. Set X k (l) = ρ k 1(RZ jk, l), and then set and Y k (l) = R( l[ρ k 1 (Z ik, l), X k (l)] ρ k 1(Z ik, l) ) Y k+1 (l) = I( l[ρ k 1 (Z ik, l), X k (l)] ρ k 1(Z ik, l) ).

13 108 BRADLEY N. CURREY Note that l[y k+1 (l), X k (l)] = l[y k+1 (l), Y k (l)] = 0. Hence if we set X k+1 (l) = ρ k (IZ j k, l) = ρ k 1 (IZ jk, l) l[ρ k 1(IZ jk, l), Y k (l)] l[x k (l), Y k (l)] X k (l) l[ρ k 1(IZ jk, l), X k (l)] l[y k (l), X k (l)] Y k (l), then l[x k+1 (l), X k (l)] = l[x k+1 (l), Y k (l)] = 0. By virtue of our assumptions for this case, l[x k+1 (l), Y k+1 (l)] does not vanish. This proves the claim. Now for this case, with e a = i k and e a+1 = i k+1 = i k, we set X k r a (l) = (l) X k+1 l[zea, X k (l)] and r a+1 (l) = (l) l[zea, X k+1 (l)]. We emphasize here that this is merely an alteration of the definitions of r a (l) and r a+1 (l) in this case. In particular the definition of ρ k (, l) is not changed. The advantage of this alteration is that it allows for the following result, which is used in the proof of Proposition (see also Proposition 2.1.1). In the remainder of this paper we shall refer to Case 0 above as Case (1.2.0), Case 1 as Case (1.2.1), and so on. Lemma Let O be a covering set in F for the fine layer e, j. Suppose that j / I, and that both j and j + 1 belong to e. Write j = e a. Then for each l O the complex numbers ζ a (l) = l[z j, r a (l)] and ζ a+1 (l) = l[z j, r a+1 (l)] are orthogonal. Proof. It suffices to show that in each of the above cases where j / I and j and j + 1 both belong to e, one has U(l) and Ũ(l) belonging to g such that l[u(l), r a+1 (l)] = l[ũ(l), r a (l)] = 0, and such that Z j = α(l) U(l) + α(l) Ũ(l) mod s j 1, where α(l) and α(l) are orthogonal complex numbers. First suppose that { j, j +1} includes a term of the index sequence j. Thus either j = j r and j + 1 = i k, or { j, j + 1} = { j r, j k }, with r < k in both cases. We set U(l) = X r (l) = R ( l[ρ r 1 (Z j, l), Y r (l)] ρ r 1 (Z j, l) ) and Then Ũ(l) = I ( l[ρ r 1 (Z jr, l), Y r (l)] ρ r 1 (Z jr, l) ). Z j = β r(l) β r (l) 2 ( U(l) + iũ(l) ) mod s j 1,

14 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 109 where β r (l) = l[ρ r 1 (Z j, l), Y r (l)]. It follows immediately that Z j = α(l)u(l) + α(l)ũ(l) mod s j 1, where α(l) and α(l) are orthogonal. If j + 1 = i k, an examination of Case (1.2.2) shows that Ũ(l) = Y k (l) mod s j 1, while if { j, j + 1} = { j r, j k }, a computation exactly as in Case (1.2.5a) shows that Ũ(l) is a real multiple of X k (l). Hence in either case we have l[u(l), r a+1 (l)] = l[ũ(l), r a (l)] = 0. Secondly, suppose that j = i k and j + 1 = i k+1. If Z jk+1 = Z jk, we set U(l) = Y k (l) and Ũ(l) = Y k+1 (l). From the definitions of Y k (l) and Y k+1 (l) we have Z j = β k(l) β k (l) 2 ( U(l) + iũ(l) ) mod s j 1, where now β k (l) = l[ρ k 1 (Z j, l), X k (l)]. and (with the alternate definitions of r a and r a+1 ) we find that l[u(l), r a+1 (l)] = l[ũ(l), r a (l)] = 0. Finally, if j = i k, j + 1 = i k+1, and Z jk+1 = Z jk, then we set U(l) = Y k (l) and Ũ(l) = Y k+1 (l). From the definitions of Y k and Y k+1 in this case we have Z j = β 1,k(l) + iβ 2,k (l) β 1,k (l) 2 + β 2,k (l) 2 ( U(l) + iũ(l) ) mod s j 1. As in the previous cases we find that U and Ũ satisfy the desired conditions. This completes the proof. For l O and t, set g a (t, l) = exp(tr a (l)) and set g a (t, l) = g 1 (t 1, l)g 2 (t 2, l) g a (t a, l) for t 2d, with g(t, l) = g 2d (t, l). Then, for each l O, Q(t, l) = g(t, l)l = n j=1 Q j (t, l)z j defines a diffeomorphism of 2d onto the coadjoint orbit of l. Note that for each 1 j n, if 1 b 2d is defined by e b j < e b+1, then Q j (t, l) = ( g b (t, l)l ) j, so that Q j (, l) depends only upon t 1, t 2,..., t b. Note also that if j / I, then Q j+1 = Q j.

15 110 BRADLEY N. CURREY 1.3. A closer look at parametrization. The form of the functions Q j (t, l) as functions of t 2d is well-known. We wish to closely examine these functions not just as functions of t, but as functions of l 1, l 2,..., l n as well. We assume that we have fixed a layer e, j belonging to the fine stratification, with all associated objects as described in the preceding section. We begin with some observations that follow immediately from the results of [Currey 1992]. Remark The definition of ρ k implies that ρ k (ρ r (Z, l), l) = ρ k (Z, l) for 0 r k, Z s, l. Remark Because X k (l) and Y k (l) are in the image of ρ k 1 (, l), we have l([v, Y k (l)]) = l([ρ k 1 (V, l), Y k (l)]), l([v, X k (l)]) = l([ρ k 1 (V, l), X k (l)]), for any V s, by the definition of ρ k 1 (, l). Formula (1.2.1) can be simplified accordingly. Remark Fix 1 k d and let Z s. Then ρ k 1 (Z, l) belongs to s l i k. Lemma Fix a covering set O F, let Y k and X k be the functions described in Lemma and let 1 k d. (i) One has X k (l) = a 1,k (l) ρ k 1 (RZ jk, l) + a 2,k (l) ρ k 1 (IZ jk, l), Y k (l) = b 1,k (l) ρ k 1 (RZ ik, l) + b 2,k (l) ρ k 1 (IZ ik, l) where a 1,k (l), a 2,k (l), b 1,k (l), and b 2,k (l) all depend only upon l 1,..., l ik. Moreover, if Case (1.2.4a) holds for k, the above statement also holds for the functions X k, Y k, X k+1, and Y k+1. (ii) Fix j such that 1 j n, and let Z s j, V s. Then l l[z, ρ k (V, l)] depends only on l 1, l 2,..., l j. Proof. We proceed by induction on k; suppose that k = 1. Note that s i1 1 = s i1 1. An examination of the construction of Y 1 (l) and X 1 (l) in [Currey 1992, Proof of Lemma 1.3], and outlined in the various cases of Section 1.2, shows that (i) is true. In fact, the functions a 1,1 (l), a 2,1 (l), b 1,1 (l), and b 2,1 (l) depend only upon the expressions l[rz j1, RZ i1 ], l[rz j1, IZ i1 ], l[iz j1, RZ i1 ], l[iz j1, IZ i1 ]. We now turn to the statement (ii) when k = 1. Observe first that, having verified (i) for k = 1, and referring to Lemma 1.2.1(v), we see that the function l l[x 1 (l), Y 1 (l)]

16 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 111 depends only on l 1, l 2,..., l i1. Now consider the function l l[z, ρ 1 (V, l)], where Z s j and V is any element of s. We have l[z, ρ 1 (V, l)] = l[z, V ] l[v, X 1(l)] l[y 1 (l), X 1 (l)] l[z, Y 1(l)] l[v, Y 1(l)] l[x 1 (l), Y 1 (l)] l[z, X 1(l)]. If j i 1, then l[z, Y 1(l)] and l[z, X 1 (l)] are both zero, whence l[z, ρ 1 (V, l)] = l[z, V ] and the conclusion follows. If j > i 1 but j j 1, then l[z, Y 1(l)] = 0, so l[z, ρ 1 (V, l)] = l[z, V ] l[v, Y 1(l)] l[x 1 (l), Y 1 (l)] l[z, X 1(l)]. Again using parts (i) and (v) of Lemma 1.2.1, we have that l l[v, Y 1 (l)] and l l[z, X 1 (l)] depend only on l 1, l 2,..., l j, and the result follows. Finally, if j > j 1, using (i) and Lemma in a similar way, we find that each factor in each term of the above depends only on l 1, l 2,..., l j. This completes the case k = 1. Now suppose that k > 1 and that (i) and (ii) hold for all 1 r k 1. We note that the induction hypothesis (together with the properties of the functions ρ r (, l)) implies that for each 1 r k 1 and 1 s k, the function l l[ρ r (RZ js, l), ρ r (RZ is, l)] = l[ρ r (RZ js, l), RZ is ] depends only upon l 1,..., l ik. (Recall here that i s i k.) Similarly, the expressions (1.3.1) l[ρ r (RZ js, l), ρ r (IZ is, l)], l[ρ r (IZ js, l), ρ r (RZ is, l)] and l[ρ r (IZ js, l), ρ r (IZ is, l)] depend only upon l 1,..., l ik. To see that (i) holds for k, we begin by observing that if (i) is true for Y k (l), it is true for X k (l) as well, by virtue of the formula X k (l) = R ( l[ρ k 1 (Z jk, l), Y k (l)] ρ k 1 (Z jk, l) ) = l[ρ k 1 (RZ jk, l), Y k (l)] ρ k 1 (RZ jk, l) +l[ρ k 1 (IZ jk, l), Y k (l)] ρ k 1 (IZ jk, l). As for Y k (l), we examine each of the five cases outlined in Section 1.2 for the formulae by which Y k (l) is defined. In Case (1.2.0), b 1,k (l) = 1 and b 2,k (l) = i, while in Case (1.2.1), b 1,k (l) = φ t,k (l) 1 β 1,k (l) and b 2,k (l) = φ t,k (l) 1 β 2,k (l) are easily seen to depend upon the expressions (1.3.1), with r = k 1. Suppose that

17 112 BRADLEY N. CURREY we are in Case (1.2.2), which means that we have r < k such that j r = i k 1 / I and Z ik = Z jr. The formula for Y k (l) in this case is ρ k 1 ( I ( l[ρr 1 (Z jr,l),y r (l)]ρ r 1 (Z jr,l) ),l ) = l[ρ r 1 (RZ jr,l),y r (l)]ρ k 1 (IZ jr,l) l[ρ r 1 (IZ jr,l),y r (l)]ρ k 1 (RZ jr,l) = l[ρ r 1 (IZ jr,l),y r (l)]ρ k 1 (RZ ik,l) l[ρ r 1 (RZ jr,l),y r (l)]ρ k 1 (IZ ik,l), where we have used Remark So b 1,k (l) = l[ρ r 1 (IZ ik, l), Y r (l)] and b 2,k (l) = l[ρ r 1 (RZ ik, l), Y r (l)] are seen to depend only upon the expressions (1.3.1). Cases (1.2.3), (1.2.4a), and (1.2.5a), are trivial: b t,k (l) = 0 or ±1, and Cases (1.2.4b) and (1.2.5b) are similar to Cases (1.2.1) and (1.2.2), respectively. Finally, in Case (1.2.4a), the definitions of X k (l), X k+1 (l), Y k (l), and Y k+1 (l) resemble those for X k (l), X k+1 (l), Y k (l), and Y k+1 (l), except with the letters X and Y interchanged, and we leave it to the reader to check that they also satisfy (i). This completes the induction step for statement (i). Turning to the statement (ii), we argue as we did for k = 1. We observe using (i) and Lemma 1.2.1(v) that the function l l[x k (l), Y k (l)] depends entirely upon the expressions (1.3.1) with r = k 1, and hence only upon l 1, l 2,..., l ik. Let Z s j and let V be any element of s. From the simplified form of (1.2.1) (Remark 1.3.2), we have l[z, ρ k (V, l)] = l[z, ρ k 1 (V, l)] l[v, X k(l)] l[y k (l), X k (l)] l[z, Y k(l)] l[v, Y k(l)] l[x k (l), Y k (l)] l[z, X k(l)]. If j i k, then l[z, Y k(l)] and l[z, X k (l)] are both zero, whence l[z, ρ k (V, l)] = l[z, ρ k 1 (V, l)] and the conclusion follows by induction. If j > i k but j j k, then l[z, Y k (l)] = 0, so l[z, ρ k (V, l)] = l[z, ρ k 1 (V, l)] Again using (i) and Lemma 1.2.1(v), we see that l[v, Y k(l)] l[x k (l), Y k (l)] l[z, X k(l)]. l l[v, Y k (l)] and l l[z, X k (l)] depend only on l 1, l 2,..., l j, and the result follows. Finally if j > j k, using (i) and Lemma in a similar way, we find that each factor in each term of the above depends only on l 1, l 2,..., l j. This completes the induction step for part (ii).

18 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 113 Lemma Assume given: (a) an index j, 1 j n such that j 1 I ; (b) indices 1 k 1, k 2,..., k p d and 1 e a1 e a2 e ap j such that e as is equal to one of i ks or j ks, 1 s p; (c) for each 1 s p, an element V s s such that ρ ks 1(V s, l) belongs to s l e as for every l ; (d) an element Z s j. Then the function l l [ [ [[Z, ρ k1 1(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] depends only on l 1, l 2,..., l j. Proof. We proceed by induction on N = p s=1 k s; if N = 1 then p = 1 and k 1 = 1, and the result is obvious. Assume that N > 1. It is clear that we may assume that k 1 > 1, and by Lemma 1.3.4, we may assume that p > 1. Note also that Y k1 1(l) s i s e k 1 1 as for all 2 s p. By the assumption about the elements V s we have l [ [ [[Z, Y k1 1(l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] = 0, and hence, for each l, (1.3.2) l [ [ [[Z, ρ k1 1(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] where Now the data = l [ [ [[Z, ρ k1 2(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] b(l) l [ [ [[Z, X k1 1(l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ], b(l) = l[ρ k 1 2(V 1, l), Y k1 1(l)]. l[x k1 1(l), Y k1 1(l)] 1 k 1 1, k 2,..., k p d, i k1 1 < e a2 < < e ap, V 1, V 2,..., V p satisfy the conditions of the lemma since ρ k1 2(V 1, l) belongs to s l i k 1 1. Hence by induction the first term of the right-hand side above, namely, l [ [ [[Z, ρ k1 2(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] depends only on l 1, l 2,..., l j. As for the second term of (1.3.2), we apply the formulas Lemma 1.3.4(i): Y k1 1(l) = b 1 (l)ρ k1 2(RZ ik1 1, l) + b 2(l)ρ k1 2(IZ ik1 1, l), X k1 1(l) = a 1 (l)ρ k1 2(RZ jk1 1, l) + a 2(l)ρ k1 2(IZ jk1 1, l),

19 114 BRADLEY N. CURREY with a 1 (l), a 2 (l), b 1 (l) and b 2 (l) depending only upon l 1, l 2,..., l ik1 1. From this and Lemma 1.3.4(ii) it follows that b(l) depends only upon l 1, l 2,..., l ik1 1. Moreover, we observe that the data 1 k 1 1, k 2,..., k p d, i k1 1 < e a2 < < e ap, RZ jk1 1, V 2,..., V p satisfy the conditions for this lemma and so, by induction, l [ [ [[Z, ρ k1 2(RZ jk1 1, l)], ρ k 2 1(V 2, l)], ], ρ k p 1(V p, l) ] depends only upon l 1, l 2,..., l j. Similarly, l [ [ [[Z, ρ k1 2(IZ jk1 1, l)], ρ k 2 1(V 2, l)], ], ρ k p 1(V p, l) ] depends only upon l 1, l 2,..., l j. We conclude that the second term of (1.3.2) depends only upon l 1, l 2,..., l j. This completes the proof. Proposition Fix O F, and for each l O, let Q(t, l) = g(t, l)l be defined as above. Then for each 1 j n and for each t 2d, the function l Q j (t, l) depends only on l 1, l 2,..., l j. Proof. Fix 1 j n; we may assume that j 1 I. Set a = max{1 b 2d e b j }. Note that r b (l) s l j for b > a, and hence exp(t b r b (l))l Z j = l j then. Now fix t 2d. Let q {0, 1, 2,... } a be a multi-index. With the conventions t q = t q 1 1 tq 2 2 tq a a and q! = q 1!q 2! q a!, we have where Q j (t, l) = g(t, l)l Z j = exp(t 1 r 1 (l))... exp(t a r a (l))l Z j = q {0,1,2,... } a w j (q, t, l), (1.3.3) w j (q, t, l) = tq q! (ad r 1 (l)) q 1 (ad r 2 (l)) q2 (ad r a (l)) q a l Z j. It remains to show that for each t 2d and each multi-index q, the function l w j (q, t, l) depends only on l 1, l 2,..., l j. Fix a multi-index q and write (e 1, e 1,..., e 1, e 2,..., e 2,..., e a,... e a ) = (e a1, e a2,..., e ap ), where on the left-hand side e b is listed q b times, for 1 b a. For each 1 s p, let 1 k s d be such that e as {i ks, j ks }. Note that i ks j holds for 1 s p. Writing Y k (l) = b 1,k (l) ρ k 1 (RZ ik, l) + b 2,k (l) ρ k 1 (IZ ik, l) as in Lemma 1.3.4, the functions b 1,ks and b 2,ks, for each 1 s p, depend only on l 1, l 2,..., l j. Similarly for the functions a 1,ks and a 2,ks that appear in the formula for X ks (l). Also by Lemma 1.3.4, the functions l[z j, X ks (l)] and l[z j, Y ks (l)]

20 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 115 depend only upon l 1, l 2,..., l j. (If Case (1.2.4a) holds for k s, replace X ks (l) by X k s (l) and the same statements hold.) Substituting the formula for r b (l) into (1.3.3) we obtain a function A(l) = A(l 1, l 2,..., l j ) such that w j (q 1, q 2,..., q a, t, l) = tq q! A(l) l[ [ [[Z, ρ k1 1(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ], where V s is one of RZ jks, IZ jks if e as = i ks, and one of RZ iks, IZ iks if e as = j ks. The factor l [ [ [[Z, ρ k1 1(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] appearing in the preceding satisfies the hypothesis of Lemma 1.3.5, and hence depends only upon l 1, l 2,..., l j. This completes the proof. Lemma Let 1 j n be an index with j 1 I, and let i k be a term of the index sequence i = {i 1 < i 2 < < i d } with i k < j. Then for each V s and for 0 r k, the function l γ j (ρ r (V, l)) depends only upon l 1, l 2,..., l j 1. Proof. We proceed by induction on r: if r = 0, the result is obvious. Suppose that r > 0, and assume that the result holds for r 1. We have γ j (ρ r (V, l)) = γ j (ρ r 1 (V, l)) l[ρ r 1(V, l), X r (l)] γ j (Y r (l)) l[y r (l), X r (l)] l[ρ r 1(V, l), Y r (l)] γ j (X r (l)). l[x r (l), Y r (l)] Note that i r i k < j; hence γ j (Y r (l)) = 0. If also j r j then γ j (X r (l)) = 0, so γ j (ρ r (V, l)) = γ j (ρ r 1 (V, l)), and the induction step is complete. Suppose that j r > j. It remains to check that each of the expressions l[ρ r 1 (V, l), Y r (l)], l[x r (l), Y r (l)] and γ j (X r (l)) depend only upon l 1, l 2,..., l j 1. Using formulas Lemma 1.3.4(i) for X r (l) and Y r (l), the fact that i r < j, and Lemma 1.3.4(ii), we see that both l[ρ r 1 (V,l),Y r (l)] and l[x r (l), Y r (l)] depend only upon l 1, l 2,..., l j 1. As for γ j (X r (l)), we apply the formula for X r (l) again: γ j (X r (l)) = a 1,r (l)γ j (ρ r 1 (RZ jr, l)) + a 2,r (l)γ j (ρ r 1 (IZ jr, l)) where a 1,r (l) and a 2,r (l) depend only on l 1, l 2,..., l ir. By the induction assumption, γ j (ρ r 1 (RZ jr, l)) and γ j (ρ r 1 (IZ jr, l)) depend only on l 1, l 2,..., l j 1. This completes the induction step and the proof. We now recall the procedure of substitution [Currey 1992, Proposition 2.6] by which Q(t, l) is simplified to obtain a map P(z, l). Let e, j be a layer

21 116 BRADLEY N. CURREY belonging to the ultrafine stratification. Given any covering set O F, then for each l O, we make substitutions z 1 = ξ 1 (t, l), z 2 = ξ 2 (t, l),..., z 2d = ξ 2d (t, l), t R 2d, l O, that result in a simplification of the expressions Q ea (t, l), for 1 a 2d. If j =e a / ϕ and e a e, then z a = Q j (t, l) (this is always the situation in the nilpotent case.) If j = e a / ϕ and e a / e (that is, j ι), then z a = c j (t, l) R ( c j (t, l) 1 Q j (t, l) ), where c j (t, l) = sign ( µ j (g a 1 (t, l) ) ζ a (l). (Here sign w = w/ w for a nonzero complex number w.) If j = e a ϕ, then z a = µ j (g a (t, l)) q j (l) 1, where q j (l) = γ j(r a (l)) ζ a (l) is a nonvanishing, µ 1 j -relatively invariant rational function on ; see [Currey 1992, Proposition 1.8, Corollary 2.2, and the definition of on p. 256]. Solving for t a in terms of z 1, z 2,..., z a and l, we obtain inverse maps 1 (z, l), 2 (z, l),..., 2d (z, l) as described in [Currey 1992, proof of Proposition 2.6, p. 261], so that n Q( (z, l), l) = P(z, l) = P j (z, l) Z j. For each l there is a submanifold T (l) of 2d, depending only on the orbit of l, such that P(, l) is an analytic bijection of T (l) with the coadjoint orbit of l. The functions P j (z, l), for 1 j n, satisfy (i) P j (z, sl) = P j (z, l) for s G; (ii) P j (z, l) = 0 mod (z 1, z 2,..., z a ), where e a j < e a+1 ; (iii) P ea (z, l) = z a mod (z 1, z 2,..., z a 1 ), with P ea (z, l) z a unless e a ι ϕ. (In the nilpotent case, ι ϕ =.) The function P(z, l) is defined on the entire ultrafine layer, independently of the covering set O, and is a precise generalization of the map of [Pukánszky 1967, Lemma 4]. Finally, one has an analytic map z : m with z(sl) = z(l) and z(l) T (l) such that P : l P(z(l), l) maps onto an orbital cross-section. The map z(l) = (z 1 (l), z 2 (l),..., z 2d (l)) is defined as follows. If e a / ϕ, set z a (l) = 0. Suppose that j = e a ϕ. Assume that if b < a, then z b (l) is defined, and set g a 1 (l) = g a 1( 1 (z 1 (l), l),..., a 1 (z 1 (l),..., z a 1 (l), l), l ). j=1

22 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 117 Then z(l) = µ j(g a 1 (l)) q j (l) where 1 + iα j = µ j /Rµ j. Set θ j (l) = l j 1 q j (l), q j (l) µ j (g a 1 (l)) l. 1+iα j, It is also shown in [Currey 1992, Lemma 2.1] that the function θ j (l) depends only upon l 1, l 2,..., l j 1. It follows from this, from the definition of the substitutions z a = ξ a (t, l) [Currey 1992, p. 263], and from Proposition and Lemma that for each 1 a 2d, both ξ a (t, l) and a (z, l) depend only on l 1, l 2,..., l ea. Thus the following is immediate. Corollary For each 1 j n and for z fixed, P j (z, ) and P j depend only upon l 1, l 2,..., l j. We now proceed with more technical results aimed at a better understanding of the structure of as a bundle over the cross-section. If j = e a e but j / ι ϕ, we already know that P j (z, l) = z a. What is needed is a better understanding of the functions Q j (t, l), and hence the functions P j (z, l), in the cases where j ι ϕ. This will be our present focus. Lemma Let 1 j n be an index with j / I, j e, and j + 1 / e. Then, for any l e, j, (i) s l j ker(γ j ), and (ii) if j = j k, then s l i k ker(γ j ). Proof. Let 1 k d with j {i k, j k }, and fix l. From the definition of i k and j k, we have Y (l) h k 1 (l) s ik and X (l) h k 1 (l) s jk so that X (l) = Z j mod s j 1, Y (l) = Z ik mod s ik 1, and l[x (l), Y (l)] = 0. Moreover, we have Z(l) s(l), such that s j = s j + -span{z j, Z(l)}. To prove part (i), assume that j = i k. If V s l j, l[v, [X (l), Y (l)]] = 0 and l[z(l), [X (l), V ]] = 0. By the Jacobi identity it follows that l[x (l), [V, Z(l)]] = 0. Since j / I, this can only happen if γ j (V ) = 0. If j = j k, the proof is the same, with Y (l) and X (l) reversing roles. This proves part (i). Now to prove (ii), assume that j = j k and let V s l i k. Then V h k 1 (l) and so [V, X (l)] = γ j (V ) Z j + W, [V, Y (l)] = γ ik (V ) Z ik + U,

23 118 BRADLEY N. CURREY with W h k 1 (l) s j 1 and U h k 1 (l) s ik 1. Hence, from the Jacobi identity, we get 0 = l[v, [X (l), Y (l)]] = (γ j (V ) + γ ik (V ))l[x (l), Y (l)], so that γ j (V ) = γ ik (V ). (Since γ j (V ) is not real, it follows that i k i k = 2.) Now referring to the cases described in Section 1.2, the proof here branches into several cases: (a) Case (1.2.0) or Case (1.2.1) holds for k: We have [X (l), Y (l)] s i k. Since V s l i k, we may repeat the same argument given for part (i) verbatim. (b) Case (1.2.2) holds for k: Here i k 1 = j r with r k 1 and we have X r (l) s ik g and X r (l) s ik g such that s ik = s ik 2+ span{x r (l), X r (l)} and such that X r (l) / h r (l), X r (l) h r (l). Now V h r (l), so [V, h r (l)] h r (l). But if γ ik (V ) = 0, then [V, X r (l)] = a X r (l) + bx r (l) + W, where W s ik 2 and b = 0. This would mean that s jr = span{[v, X r (l)], X r (l)} + s jr h r (l) + s j r, which contradicts the definition of j r = min{1 j n h r 1 (l) s j h r (l)}. (c) Case (1.2.4) holds for k: We have X k (l) and X k (l) belonging to s j g with s j = span{x k (l), X k (l)}+s j, l[x k (l), Y k (l)] = 0, and l[ X k (l), Y k (l)] = 0. This means that X k (l) / h k (l), and X k (l) h k+1 (l), the latter because, by virtue of our assumption that j + 1 / e, we have j k+1 > j + 1. Now γ j (V ) = 0 if and only if γ j (ρ k+1 (V, l)) = 0, and ρ k+1 (V, l) belongs to h k+1 (l). Hence if γ j (ρ k+1 (V, l)) = 0, then [ρ k+1 (V, l), X k (l)] = a X k (l)+bx k (l) mod s j, where b = 0. This would imply that s j h k (l) + s j, contradicting the definition of j k = j. (d) Case (1.2.5) holds for k: This case is similar to (c), and we omit the details. Lemma Suppose given j with 1 j n and j 1 I, and k 1 k d with i k < j. Assume further that if j = j r for some r < k, then j / I and j +1 / e. Then, for 0 r k 1 and for each V s, the function l l[z j, ρ r (V, l)] defined on e, j is of the form l[z j, ρ r (V, l)] = γ j (ρ r (V, l))l j + u(l), where u(l) depends only upon l 1, l 2,..., l j 1. Proof. We proceed by induction on r, the result being clear for r = 0. Assume that r > 0 and that the result holds for r 1. This means in particular that we may assume that l[z j, ρ r 1 (V, l)] = γ j (ρ r 1 (V, l))l j + u 0 (l 1, l 2,..., l j 1 ). By our hypothesis and the properties of sequence pairs we have i r < j, and also j + 1 = j r if j / I. We therefore have three cases: j < j r, j = j r, and j j r.

24 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 119 Case 1: j < j r. Here l[z j, Y r (l)] = l[ρ r 1 (Z j, l), Y r (l)] = 0, so l[z j, ρ r (V, l)] = l[z j, ρ r 1 (V, l)] c(l)l[z j, X r (l)]. where c(l) = l[ρ r 1(V, l), Y r (l)]. l[x r (l), Y r (l)] By Lemma 1.3.4, we have Y r (l) = b r,1 (l)ρ r 1 (RZ ir, l) + b r,2 (l)ρ r 1 (IZ ir, l), X r (l) = a r,1 (l)ρ r 1 (RZ jr, l) + a r,2 (l)ρ r 1 (IZ jr, l), where a r,1 (l), a r,2 (l), b r,1 (l), and b r,2 (l) depend only upon l 1, l 2,..., l ir. It follows from these formulas and the induction hypothesis that c(l) depends only upon l 1, l 2,..., l j 1. Also by induction we have l[z j, ρ r 1 (RZ jr, l)] = γ j (ρ r 1 (RZ jr, l))l j + v 1 (l 1, l 2,..., l j 1 ), l[z j, ρ r 1 (IZ jr, l)] = γ j (ρ r 1 (IZ jr, l))l j + v 2 (l 1, l 2,..., l j 1 ), and it follows that we have u 1 (l 1, l 2,..., l j 1 ) such that Hence l[z j, X r (l)] = γ j (X r (l))l j + u 1 (l 1, l 2,..., l j 1 ). l[z j, ρ r (V, l)] = γ j (ρ r 1 (V, l))l j + u 0 (l) c(l) ( γ j (X r (l))l j + u 1 (l) ) = γ j (ρ r (V, l))l j + u(l), where u(l) = u 0 (l) c(l)u 1 (l) depends only upon l 1, l 2,..., l j 1. Case 2: j = j r. Here j / I and j + 1 / e. By Remark and Lemma 1.3.9, for each l, the image of ρ r 1 (, l) is contained in ker(γ j ). This, combined with the induction hypothesis, implies that for any V s, the expression l[z j, ρ r 1 (V, l)] depends only upon l 1, l 2,..., l j 1. Combining this with Lemma 1.3.4, we find that the expressions c(l) = l[ρ r 1(V, l), Y r (l)], d(l) = l[ρ r 1(V, l), X r (l)] l[x r (l), Y r (l)] l[y r (l), X r (l)] depend only upon l 1, l 2,..., l j 1, and hence that l[z j, ρ r (V, l)] = l[z j, ρ r 1 (V, l)] c(l) l[z j, X r (l)] d(l)l[z j, Y r (l)] depends only upon l 1, l 2,..., l j 1 also. Since ρ r (V, l) ker(γ j ), we are done with this case. Case 3: j j r. This is similar to Case 1, with an additional term that is handled in a way precisely analogous to the arguments in Case 1. We omit the details.

25 120 BRADLEY N. CURREY Lemma Assume given: (a) an index j with 1 j n and j 1 I, and such that, if j = j r, then j / I and j + 1 / e; (b) indices 1 k 1, k 2,..., k p d and 1 e a1 e a2 e ap j such that e as is equal to one of i ks or j ks, for 1 s p; (c) for each 1 s p, an element V s s such that for every l e, j, ρ ks 1(V s, l) belongs to s l e. as Then, for each l e, j, l [ [ [[Z j, ρ k1 1(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] where y(l) depends only upon l 1, l 2,..., l j 1. = p γ j (ρ ks 1(V s, l))l j + y(l), Proof. Set r s = k s 1, for 1 s p. As in Lemma 1.3.5, we proceed by induction on N = p s=1 k s, and by Lemma , we may assume that p > 1. Suppose first that r 1 = 0. Writing [Z j, V 1 ] = γ j (V 1 )Z j + W with W s j 1, we apply induction to l [ [... [Z j, ρ r2 (V 2, l)],... ], ρ r p (V p, l) ] and Lemma to obtaining that s=1 y 1 (l) = l [ [... [W, ρ r2 (V 2, l)],... ], ρ r p (V p, l) ], l [ [ [[Z j, V 1 ],ρ r2 (V 2,l)], ],ρ r p (V p,l) ] ( p ) = γ j (V 1 ) γ j (ρ rs (V s,l))l j + y 0 (l) +l [ [...[W,ρ r2 (V 2,l)],...],ρ r p (V p,l) ] = γ j (V 1 ) s=2 p γ j (ρ rs (V s,l))l j +γ j (V 1 ) y 0 (l)+ y 1 (l). s=2 Now suppose that r 1 > 0. From our assumption about the indices and the elements V s, we have l [ [... [Y r1 (l), ρ r2 (V 2, l)],... ], ρ r p (V p, l) ] = 0. Thus (1.3.4) l [ [ [[Z j, ρ r1 (V 1, l)], ρ r2 (V 2, l)], ], ρ r p (V p, l) ] = l [ [ [[Z j, ρ r1 1(V 1, l)], ρ r2 (V 2, l)], ], ρ r p (V p, l) ] c(l)l [ [ [[Z j, X r1 (l)], ρ r2 (V 2, l)], ], ρ r p (V p, l) ],

26 ORBITAL PARAMETERS AND PLANCHEREL MEASURE FOR LIE GROUPS 121 where c(l) = l[ρ r 1 1(V 1, l), Y r1 (l)]. l[x r1 (l), Y r1 (l)] We now proceed in much the same way as in the proof of Lemma Looking at the first term of the right-hand side of 1.3, we observe that the data 1 k 1 1, k 2,..., k p d, i k1 1 < e a2 < < e ap, V 1,..., V p satisfy the hypothesis of this lemma, so by induction, l [ [ [[Z j, ρ r1 1(V 1, l)], ρ r2 (V 2, l)], ], ρ r p (V p, l) ] = γ j (ρ r1 1(V 1, l)) p γ j (ρ rs (V s, l))l j + y 0 (l), where y 0 (l) depends only upon l 1, l 2,..., l j 1. Turning to the second term, we apply formulas Lemma 1.3.4(i) to conclude that c(l) depends only upon l 1, l 2,..., l ik1 1. We then observe that the data 1 k 1 1, k 2,..., k p d, i k1 1 < e a2 < < e ap, RZ jk1 1, V 2,..., V p s=2 satisfy the conditions for this lemma, and so, by induction, l [ [ [[Z j, ρ r1 1(RZ jk1 1, l)], ρ r 2 (V 2, l)], ], ρ r p (V p, l) ] = γ j (ρ r1 1(RZ jr1, l)) p γ j (ρ rs (V s, l))l j + y 1 (l), where y 1 (l) depends only upon l 1, l 2,..., l j 1. An entirely similar formula holds involving IZ jk1 1 instead of RZ j k1 1 and a remainder y 2(l) depending only upon l 1, l 2,..., l j. Using the formula for X r1 (l) from Lemma 1.3.4, we can substitute the preceding into equation 1.3 to get s=2 l [ [ [[Z j, ρ k1 1(V 1, l)], ρ k2 1(V 2, l)], ], ρ k p 1(V p, l) ] = γ j (ρ r1 1(V 1, l)) p γ j (ρ rs (V s, l))l j + y 0 (l) s=2 ( c(l)a 1 (l) γ j (ρ r1 1(RZ jr1, l)) ( c(l)a 2 (l) γ j (ρ r1 1(IZ jr1, l)) p s=2 p s=2 ) γ j (ρ rs (V s, l))l j + y 1 (l) ) γ j (ρ rs (V s, l))l j + y 2 (l)