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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión 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.
Let be a connected, simply connected, and completely solvable Lie group, with Lie algebra
. Let
be the dual of
. The equivalence classes of irreducible unitary representations
of
is parameterized by the coadjoint orbits
via the Kirillov bijective map
![K : ˆG → G *∕G](/img/revistas/ruma/v48n1/1a057x.png)
We recall that if and
, then there exists an analytic subgroup
of
and a unitary character
of
, such that the induced representation
is equivalent to
Moreover the push forward of a Plancherel measure in
is a measure equivalent to a Lebesguian measure on convenient set of representatives in
for
Let in
and set
its Fourier transform, let
and
. By Bénédicks theorem [1, Theorem 2], if
and
then
a.e. Here,
denote Lebesgue measure on
. That is, for
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
of
as well as an explicit description of Plancherel measure on
. 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 and
,
Definition 1.1. G has the qualitative uncertainty principle if and
then
m-a.e.
Remark 1.1. The group () has the qualitative uncertainty principle [1, Theorem 2].
Let be a connected, simply connected, and completely solvable Lie group, with the Lie algebra
. Let
be its dual. Since
is completely solvable, there exists a chain of ideals of
![(0) = G ⊂ G ⊂ ...⊂ G = G 0 1 n](/img/revistas/ruma/v48n1/1a0550x.png)
such that the dimension of is
, for all
. We fix an ordered basis
of
such that
is spanned by the vectors
,
Let
be the dual basis of
We fix a Lebesgue measure
on
and a right invariant Haar measure
on
such that
where
![-adX JG (X ) = |det(1 --e----)| adX](/img/revistas/ruma/v48n1/1a0566x.png)
Let be the modular function such that for all
,
. Let
be a co-adjoint orbit in
and
. The bilinear form
defines a skew-symmetric and nondegenerate bilinear form on
. Since the map
induces an isomorphism between
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](/img/revistas/ruma/v48n1/1a0583x.png)
is a canonical measure on . Lemma 3.2.2 in [2] says that there exists a nonzero rational function
on
such that
![ψ(g.l) = δ(g)-1ψ(l),g ∈ G, l ∈ G *](/img/revistas/ruma/v48n1/1a0587x.png)
and there exists a unique measure on
such that
![∫ ∫ ∫ φ(l)|ψ (l)|dl = ( φ(l)dBO (l))dm ψ (O ) G* G*∕G O](/img/revistas/ruma/v48n1/1a0590x.png)
for all Borel function on
. B.N. Currey [3,] gave an explicit description of the measure
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
, a subset
of
, a subset
of
, for each
a real valued rational function
, non vanishing on
, and real analytic functions
in the variables
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
. 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
and the coordinates for
are obtained by identifying
with
, which is the parametrization of
in the nilpotent case. If
is not empty and
the number of elements in
. From [3], for each
is a non empty Zariski open subset and
(disjoint union). Set
. from [3] we have:
![W ε = {l ∈ (WD ⊕ WM ) ∩ U | for each j = jrm ∈ M, lj = εm - pj(l1,....,lj-1)}](/img/revistas/ruma/v48n1/1a05154x.png)
is a rational nonsingular function on
.
Let From [3], there is a Zariski open subset
of
and a rational function
such that
is the graph of
. From [3], the projection of
into
parallel to
defines a diffeomorphism of
with
.
Summarizing: let be connected, simply connected and completely solvable Lie group. Let
be a Jordan-Holder basis of
. Then, there is a finite family of disjoint open subsets
of
and there is a subspace
of
such that for each
, the orbits in
are parameterized by a Zariski open subset
of
The union of this open sets determines an open dense subset of
whose complement has Plancherel measure zero.
Consider the group
![{ ( ) } G = a b | a > 0,b ∈ ℝ 0 1](/img/revistas/ruma/v48n1/1a05181x.png)
We use the notation
Matrix multiplication is:
![(a1,b1)(a2,b2) = (a1a2, a1b2 + b1)](/img/revistas/ruma/v48n1/1a05183x.png)
and the inverse is
![-1 -1 -1 (a,b) = (a ,- ba ).](/img/revistas/ruma/v48n1/1a05184x.png)
The Lie algebra of
is the set of matrices
![{( ) } G = x y x, y ∈ ℝ 0 0](/img/revistas/ruma/v48n1/1a05187x.png)
We choose as ordered base
![( ) ( ) 1 0 0 1 X = 0 0 and Y = 0 0](/img/revistas/ruma/v48n1/1a05189x.png)
We have . Thus the group is not nilpotent.
Let the dual basis of
. Let
The orbits of
in
are: the upper half plane
, the lower half plan
and the points
. Here,
and
, so that
and
is spanned by the vector
. The Zariski open sets
and
are the half planes of
and
. Since there are two orbits, the set
![W = {l ∈ WM ∩ U : | qj2(l) |= 1,j2 ∈ M }](/img/revistas/ruma/v48n1/1a05209x.png)
has exactly two points. We have and
. The Zariski open set
or
of
, reduces to a point.
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
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
.
Theorem 5.1. Let be a connected, simply connected, completely solvable Lie group with the unitary dual
, and let
be integrable function on
. If
and
then
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
(see [1,theorem2]). Assume that the result is true for all completely solvable Lie groups of dimension
. Suppose that
are finite. From [4, lemma 1.6],
is finite. To conclude, it remains to show that
is finite. We can assume that
is contained in
(It suffices to take
as the finite union of
).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 group has the qualitative uncertainty principle.
Question 5.1. Do the exponential solvable Lie groups have the qualitative uncertainty principle ?
[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