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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007

 

A qualitative uncertainty principle for completely solvable Lie groups

B. Bouali

Abstract. In this paper, we study a qualitative uncertainty principle for completely solvable Lie groups.

2000 Mathematics Subject Classification. 22E25, 22E27.

Keywords and phrases. Completely solvable Lie groups, uncertainty principle, orbit method.

1. Introduction

Let G be a connected, simply connected, and completely solvable Lie group, with Lie algebra G . Let  * G be the dual of G . The equivalence classes of irreducible unitary representations ˆG of G is parameterized by the coadjoint orbits  * G ∕G via the Kirillov bijective map

K : ˆG → G *∕G

We recall that if (Vρ,ρ) ∈ Gˆ and l ∈ K (ρ) , then there exists an analytic subgroup H of G and a unitary character ξ of H , such that the induced representation ρ is equivalent to IndG ξ. H Moreover the push forward of a Plancherel measure in Gˆ is a measure equivalent to a Lebesguian measure on convenient set of representatives in G * for  ˆ G.

Let f in L1(ℝn ) and set ˆf its Fourier transform, let Af = {x ∈ ℝn : f (x) ⁄= 0} and B = {x ∈ ℝn : ˆf(x) ⁄= 0} f . By Bénédicks theorem [1, Theorem 2], if λ(A ) < ∞ f and λ (Bf ) < ∞ then f = 0 a.e. Here, λ denote Lebesgue measure on  n ℝ . That is, for  n ℝ the qualitative uncertainty principle holds.

In this note we prove that a completely solvable Lie group has the qualitative uncertainty principle. In [4] we showed the theorem for nilpotent Lie groups, by induction on the dimension of G . To prove the theorem we apply induction, for this, we need an explicit description of the dual space  ˆ G of G as well as an explicit description of Plancherel measure on ˆ G . For our approach we use a result of B.N. Currey [3], which is a generalization of a result of L. Pukanszky. Let G be a locally compact group. Denote a fixed Haar measure on G by m and the corresponding Plancherel measure on ˆG by μ.

Let Af = {x ∈ G : f(x) ⁄= 0} and  ˆ ˆ Bf = {π ∈ G : f(π) ⁄= 0} ,

Definition 1.1. G has the qualitative uncertainty principle if m (Af ) < ∞ and μ(Bf ) < ∞, then f = 0 m-a.e.

Remark 1.1. The group ( n ℝ ,+ ) has the qualitative uncertainty principle [1, Theorem 2].

2. Preliminaries

Let G be a connected, simply connected, and completely solvable Lie group, with the Lie algebra G . Let  * G be its dual. Since G is completely solvable, there exists a chain of ideals of G

(0) = G ⊂ G ⊂ ...⊂ G = G 0 1 n

such that the dimension of Gj is j , for all j ≤ n . We fix an ordered basis B = {X1, X2, ..,Xn } of G such that Gj is spanned by the vectors X1, ..,Xj ,1 ≤ j ≤ n. Let B * = {X *1,X *2,..,X *n} be the dual basis of B. We fix a Lebesgue measure dX on G and a right invariant Haar measure m on G such that m (expX ) = JG (X )dX where

 -adX JG (X ) = |det(1 --e----)| adX

Let δ be the modular function such that for all g ∈ G ,  ′ ′ m (gg ) = δ(g)m(g ) . Let O be a co-adjoint orbit in  * G and l ∈ O . The bilinear form Bl: (X, Y ) → l([X, Y]) defines a skew-symmetric and nondegenerate bilinear form on  l G ∕G . Since the map X → X.l induces an isomorphism between G∕Gl and the tangent space of O at l , the bilinear form Bl defines a nondegenerate 2-form wl on this tangent space. If 2k is the dimension of O we note that

B := (2k )-k(k!)-1w ∧ w ∧ ..∧ w (k times ) O l l l

is a canonical measure on O . Lemma 3.2.2 in [2] says that there exists a nonzero rational function ψ on G* such that

ψ(g.l) = δ(g)-1ψ(l),g ∈ G, l ∈ G *

and there exists a unique measure m ψ on  * G ∕G such that

∫ ∫ ∫ φ(l)|ψ (l)|dl = ( φ(l)dBO (l))dm ψ (O ) G* G*∕G O

for all Borel function φ on G* . B.N. Currey [3,] gave an explicit description of the measure m ψ with the help of the coadjoint orbits. We recall the theorem proved by B.N. Currey which is a essential tool to prove our main theorem.

Theorem 2.1. Let G be a connected, simply connected and completely solvable Lie group. There exists a Zariski open subset U in G * , a subset J = {j1 < j2 < ...< j2k} of {1, 2,...,n} , a subset M = {jr < jr < ...< jr } 1 2 a of J , for each j ∈ M a real valued rational function qj , non vanishing on U , and real analytic functions Pj in the variables w1, w2,..,w2k,l1,l2,..,ln such that the following hold.

  1. If a denotes the number of elements of M , for each ε ∈ {1, - 1 }a , the set

    U ε = {l ∈ U | sign of qjrm (l) = εm, 1 ≤ m ≤ a}

    is a non empty open subset in G * .

  2. Define V ⊂ ℝ2k by V = ∏ Rr , where Rr = ]0,∞ [ if jr ∈ M and Rr = ℝ otherwise. Let ε ∈ {1,- 1}a, for v ∈ V , define εv ∈ ℝ2k by (εv) = ε v j m j if j = jrm ∈ M and (εv )j = vj otherwise. Then for each l ∈ U ε , the mapping  ∑ * v - → j∈J Pj (εv, l)X j is a diffeomorphism of V with the coadjoint orbit of l .
  3. Define WD as the subspace spanned by the vectors {X * | i ⁄∈ J} i and WM the subspace spanned by the vectors  * {X i | i ∈ M } Then the set

    W = {l ∈ (WD ⊕ WM ) ∩ U || qj(l) |= 1,j ∈ M }

    is a cross-section for the coadjoint orbits in U . for each j ∈ M the rational function qj is of the form qj(l) = lj + pj(l1,l2,..,lj-1) , where pj is a rational function.
  4. For each l ∈ U , let  a ε(l) ∈ {1,- 1} such that l ∈ Uε(l) . Then the mapping P : V × W - → U , defined by  ∑ P (v,l) = j Pj (ε(l)v,l)Xj* , is a diffeomorphism.

If the subset M is empty, then W = WD ∩ U and the coordinates for W are obtained by identifying WD with  n-2k ℝ , which is the parametrization of ˆG in the nilpotent case. If M is not empty and a the number of elements in M . From [3], for each ε ∈ {1,- 1}a U ε is a non empty Zariski open subset and U = ∪εUε (disjoint union). Set W ε = W ∩ U ε . from [3] we have:

W ε = {l ∈ (WD ⊕ WM ) ∩ U | for each j = jrm ∈ M, lj = εm - pj(l1,....,lj-1)}

pj is a rational nonsingular function on U .

Let  a ε ∈ {- 1,1} . From [3], there is a Zariski open subset Λ ε of WD and a rational function p ε: Λε -→ WM such that W ε is the graph of pε . From [3], the projection of U ε into WD parallel to WJ defines a diffeomorphism of WD with Λ ε .

Summarizing: let G be connected, simply connected and completely solvable Lie group. Let {X *,X *,...,X *} 1 2 n be a Jordan-Holder basis of G * . Then, there is a finite family of disjoint open subsets U ε of  * G and there is a subspace WD of  * G such that for each ε , the orbits in Uε are parameterized by a Zariski open subset Λ ε of WD. The union of this open sets determines an open dense subset of  * G ∕G whose complement has Plancherel measure zero.

3. The ax + b Group.

Consider the group

 { ( ) } G = a b | a > 0,b ∈ ℝ 0 1

We use the notation

 ( ) a b (a, b) = 0 1 . Matrix multiplication is:

(a1,b1)(a2,b2) = (a1a2, a1b2 + b1)

and the inverse is

 -1 -1 -1 (a,b) = (a ,- ba ).

The Lie algebra G of G is the set of matrices

 {( ) } G = x y x, y ∈ ℝ 0 0

We choose as ordered base B,

 ( ) ( ) 1 0 0 1 X = 0 0 and Y = 0 0

We have [X, Y ] = Y . Thus the group is not nilpotent.

Let  * * {X ,Y } the dual basis of  * G . Let  * * l = αX + βY ∈ G. The orbits of G in  * G are: the upper half plane β > 0 , the lower half plan β < 0 and the points (α, 0) . Here, J = {j1,j2} = {1,2} and M = {j2} ⊂ J , so that V = ℝ2, V = ]0, +∞ [× ℝ. W = (0) D and W M is spanned by the vector X * j2 . The Zariski open sets U + and U - are the half planes of  * G and U = U+ ∪ U - . Since there are two orbits, the set

W = {l ∈ WM ∩ U : | qj2(l) |= 1,j2 ∈ M }

has exactly two points. We have W+ = W ∩ U+ and W - = W ∩ U- . The Zariski open set Λ+ or Λ - of WD , reduces to a point.

4. Fourier transform.

We must consider two cases(see [5]):

  1. All the orbits in general position are saturated with respect to Gn-1. That is, for each l ∈ G *, Gl ⊂ G . n-1 Then, we may and will choose a basis of G

    BW ε = {X1 (l),X2 (l),....,Xn -1(l),Xn }.

    where the last vector of the basis does not depend on l . We apply the previous setting to G := exp (G ). n- 1 n-1 Let J = J \ {n,j } 1 1 the index set for G n-1 , then M1 is a subset of J1, let a1 denote the number elements of M1 . For each  a ε1 ∈ {- 1,1} 1 , the set Uε1 is nonempty open subset of  * Gn- 1 . Let WD1 = WD ⊕ ℝX *j1 and then WM1 is the subspace spanned by X *j,j ∈ M1 . We apply the inductive hypothesis to Gn- 1 , hence, there is a Zariski open subset Λε ⊂ WD 1 1 and a rational function pε : Λε → WD 1 1 1 such that W ε 1 is the graph of p ε1 . Let Λ ε1 denote the projection of Λ ε on G * n- 1 . From [5,lemma 3.2], the measure dμ1 on W ε1 in terms of the measure dμ on W ε and  * dX j1 is  * dμ1 = dμ × dX j1

  2. If some orbit G ⋅ l in general position is not saturated with respect to Gn -1 , we can still obtain a basis of G such that the last vector of the basis does not depend on l ,  l Xn ∈ G and  lj Xi ∈ Gj for certain j with lj = l | Gj . In this case since Gl = Gln-1 , we have WD = WD1 + ℝXn . Moreover Λ ε = Λ ε1 + ℝXn * . The Plancherel measure can be written as d μ(l) = d μ1 × dX *n .

5. The main theorem.

Theorem 5.1. Let G be a connected, simply connected, completely solvable Lie group with the unitary dual Gˆ , and let f be integrable function on G (f ∈ L1 (G)) . If m (Af ) < ∞ and μ(Bf ) < ∞ then f = 0 almost every where.

Proof 5.1. We proceed by induction on the dimension n of G . The result is true if the dimension of G is one, since G ~= ℝ (see [1,theorem2]). Assume that the result is true for all completely solvable Lie groups of dimension n - 1 . Suppose that m (Af ),μ(Bf ) are finite. From [4, lemma 1.6], m1(Aft ) is finite. To conclude, it remains to show that μ (B t) 1 f is finite. We can assume that B f is contained in W ε (It suffices to take Bf as the finite union of Bf ∩ W ε ).We consider tho cases

  1. We suppose that  l G ⊂ Gn -1 for all l ∈ W ε. That is, all the orbits in general position are saturated with respect to Gn-1 . For  * φ ∈ G , let φ0 be the restriction of φ to Gn -1 , then  G πφ = Ind Gn-1πφ0 is irreducible. From [6, proposition 2.5] we have:

    ∫ ⊕ ∫ ⊕ IndGG πφ0dλG* (φ0) ≃ πφdλG*(φ ) (1) G*n- 1 n-1 n-1 G*

    where dλG* is the Lebesgue measure on G* and dλG*n-1 is the Lebesgue measure on G * n-1 . From the formula (1) and the definition of Xn, we conclude that the map φ → φ 0 is an isomorphism which respect to the measures dλ * Gn-1 and dλ * G , then

    μ1 (Bft ) = μ(Bf ) < ∞.

    By induction hypothesis f t = 0 almost everywhere on Gn- 1 for almost everywhere t ∈ ℝ, which implies that f = 0 almost everywhere on G by using the theorem of Fubini.

  2. Some orbit G ⋅ l is not satured with respect to Gn- 1. That is, Gl ⁄⊂ Gn- 1 for some l ∈ W ε . For φ0 ∈ Gn*- 1 , we choose an extension φ defined by φ(Xn ) = 0 . From this we have

     ∫ ⊕ indGGn-1πφ0 ~ πφ0+sX*nds. ℝ

    Hence

     ∫ μ(Bf ) = μ1(Bft)dt < ∞. ℝ

    Then for almost everywhere t ∈ ℝ , μ1(Bft) is finite. By inductive hypothesis f t = 0 almost everywhere on Gn- 1 for almost everywhere t in ℝ , which implies that f = 0 almost everywhere on G by Fubini's theorem.

Remark 5.1. The ax + b group has the qualitative uncertainty principle.

Question 5.1. Do the exponential solvable Lie groups have the qualitative uncertainty principle ?

References

[1]    M. Benedicks, On Fourier transform of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl. 106 (1985), no. 1, 180-183.        [ Links ]

[2]    A. Kleppner and Lipsman, The Plancherel formula for groups extensions, Ann. Sci. cole Norm. Sup. (4) 5 (1972), 459-516.        [ Links ]

[3]    B.N.Currey, An explicit Plancherel formula for completely solvable Lie groups, Michigan Math Journal,38(1991) 75-87.        [ Links ]

[4]    B. Bouali and M. Hemdaoui, Principe d'incertitude qualitatif pour les groupes de Lie nilpotents, Revista Matematica complutense, 2004,17,Num. 2, 277-285.        [ Links ]

[5]    G.Garimella, Weak Paley-Wiener property for completely solvable Lie groups, Pacific Journal of mathematics, vol 187 No 1,(1999)51-63.        [ Links ]

[6]    A. Baklouti, J. Ludwig, Désintégration des représentations monômiales des groupes de Lie nilpotents, Journal of Lie theory,volume 9(1999) 157-191.        [ Links ]

B. Bouali
University of Mohammed premier,
OUJDA, MOROCCO

Recibido: 13 de febrero de 2006
Aceptado: 20 de julio de 2007