A qualitative uncertainty principle for completely solvable Lie groups

Bouali, B.

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Revista de la Unión Matemática Argentina

Print version ISSN 0041-6932On-line version ISSN 1669-9637

Rev. Unión Mat. Argent. vol.48 no.1 Bahía Blanca Jan./June 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 be a connected, simply connected, and completely solvable Lie group, with Lie algebra . Let * G be the dual of . The equivalence classes of irreducible unitary representations of is parameterized by the coadjoint orbits * G ∕G via the Kirillov bijective map

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

Let in L1(ℝn ) and set 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, 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 . To prove the theorem we apply induction, for this, we need an explicit description of the dual space ˆ G of 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 be a locally compact group. Denote a fixed Haar measure on by and the corresponding Plancherel measure on by

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

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

Let be the modular function such that for all g ∈ G , ′ ′ m (gg ) = δ(g)m(g ) . Let 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 at , the bilinear form defines a nondegenerate 2-form on this tangent space. If is the dimension of we note that

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

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

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

If denotes the number of elements of , for each , the set

is a non empty open subset in .
Define by , where if and otherwise. Let for , define by if and otherwise. Then for each , the mapping is a diffeomorphism of with the coadjoint orbit of .
Define as the subspace spanned by the vectors and the subspace spanned by the vectors Then the set
is a cross-section for the coadjoint orbits in . for each the rational function is of the form , where is a rational function.
For each , let such that . Then the mapping , defined by , is a diffeomorphism.

If the subset is empty, then W = WD ∩ U and the coordinates for are obtained by identifying with n-2k ℝ , which is the parametrization of in the nilpotent case. If is not empty and the number of elements in . 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)}

is a rational nonsingular function on .

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

Summarizing: let 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 of * G such that for each , the orbits in 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 Group.

Consider the group

We use the notation

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

and the inverse is

The Lie algebra of is the set of matrices

We choose as ordered base

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 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 or Λ - of , reduces to a point.

4. Fourier transform.

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

All the orbits in general position are saturated with respect to That is, for each Then, we may and will choose a basis of

where the last vector of the basis does not depend on . We apply the previous setting to Let $J = J \ {n,j } 1 1$ the index set for , then is a subset of let denote the number elements of . For each , the set is nonempty open subset of . Let and then is the subspace spanned by . We apply the inductive hypothesis to , hence, there is a Zariski open subset and a rational function such that is the graph of . Let denote the projection of on . From [5,lemma 3.2], the measure on in terms of the measure on and is
If some orbit in general position is not saturated with respect to , we can still obtain a basis of such that the last vector of the basis does not depend on , and for certain with . In this case since , we have . Moreover . The Plancherel measure can be written as .

5. The main theorem.

Theorem 5.1. Let be a connected, simply connected, completely solvable Lie group with the unitary dual , and let be integrable function on (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 of . The result is true if the dimension of 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 as the finite union of Bf ∩ W ε ).We consider tho cases

We suppose that for all That is, all the orbits in general position are saturated with respect to . For , let be the restriction of to , then is irreducible. From [6, proposition 2.5] we have:

where is the Lebesgue measure on and is the Lebesgue measure on . From the formula (1) and the definition of we conclude that the map is an isomorphism which respect to the measures and , then

By induction hypothesis almost everywhere on for almost everywhere which implies that almost everywhere on by using the theorem of Fubini.
Some orbit is not satured with respect to That is, for some . For , we choose an extension defined by . From this we have

Hence

Then for almost everywhere , is finite. By inductive hypothesis almost everywhere on for almost everywhere in , which implies that almost everywhere on 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