Another proof for the continuity of the Lipsman mapping
UDC 515.1 We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle , \rangle$. By $\hat G $ we denote the unitary dual of $G$ and by ${\mathfrak{g}^{ \...
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507043893346304 |
|---|---|
| author | Messaoud, A. Rahali, A. Messaoud, A. Rahali, A. |
| author_facet | Messaoud, A. Rahali, A. Messaoud, A. Rahali, A. |
| author_sort | Messaoud, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:52Z |
| description | UDC 515.1
We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle , \rangle$. By $\hat G $ we denote the unitary dual of $G$ and by ${\mathfrak{g}^{ \ddagger} /} G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G$. It was pointed out by Lipsman that the correspondence between $\hat{G} $ and ${\mathfrak{g}^{ \ddagger} /} G$ is bijective. Under some assumption on $G$, we give another proof for the continuity of the orbit mapping (Lipsman mapping)$$\Theta : {\mathfrak{g}^{ \ddagger} /} G - \rightarrow \hat{G} .$$ |
| doi_str_mv | 10.37863/umzh.v72i7.548 |
| first_indexed | 2026-03-24T02:03:03Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i7.548
UDC 515.1
A. Messaoud, A. Rahali (Univ. Sfax, Tunisia)
ANOTHER PROOF FOR THE CONTINUITY OF THE LIPSMAN MAPPING
ЩЕ ОДНЕ ДОВЕДЕННЯ НЕПЕРЕРВНОСТI ВIДОБРАЖЕННЯ ЛIПСМАНА
We consider the semidirect product G = K \ltimes V where K is a connected compact Lie group acting by automorphisms on
a finite dimensional real vector space V equipped with an inner product \langle , \rangle . By \widehat G we denote the unitary dual of G and
by g\ddagger /G the space of admissible coadjoint orbits, where g is the Lie algebra of G. It was pointed out by Lipsman that the
correspondence between \widehat G and g\ddagger /G is bijective. Under some assumption on G, we give another proof for the continuity
of the orbit mapping (Lipsman mapping)
\Theta : g\ddagger /G - \rightarrow \widehat G.
Розглядається напiвпрямий добуток G = K \ltimes V, де K — зв’язна компактна група Лi автоморфiзмiв, що дiють на
скiнченновимiрному дiйсному векторному просторi V iз внутрiшнiм добутком \langle , \rangle . Нехай \widehat G — унiтарний дуал G, а
g\ddagger /G — простiр допустимих коспряжених орбiт, де g — алгебра Лi для G. Лiпсман зазначив, що вiдповiднiсть мiж\widehat G та g\ddagger /G є бiєкцiєю. При деяких припущеннях на G ми пропонуємо нове доведення неперервностi вiдображення
орбiт (вiдображення Лiпсмана)
\Theta : g\ddagger /G - \rightarrow \widehat G.
1. Introduction. Let G be a second countable locally compact group and \widehat G the unitary dual of G,
i.e., the set of all equivalence classes of irreducible unitary representations of G. It is well-known
that \widehat G equipped with the Fell topology [6]. The description of the dual topology is a good candidate
for some aspects of harmonic analysis on G (see, for example, [4, 5]). For a simply connected
nilpotent Lie group and more generally for an exponential solvable Lie group G = \mathrm{e}\mathrm{x}\mathrm{p}(g), its dual
space \widehat G is homeomorphic to the space of coadjoint orbits g\ast /G through the Kirillov mapping (see
[8]). In the context of semidirect products G = K \ltimes N of compact connected Lie group K acting
on simply connected nilpotent Lie group N, then it was pointed out by Lipsman in [9], that we have
again an orbit picture of the dual space of G. The unitary dual space of Euclidean motion groups is
homeomorphic to the admissible coadjoint orbits [5]. This result was generalized in [4], for a class
of Cartan motion groups.
In this paper, we consider the semidirect product G = K \ltimes V, where K is a connected compact
Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an
inner product \langle , \rangle . In the spirit of the orbit method due to Kirillov, R. Lipsman established a bijection
between a class of coadjoint orbits of G and the unitary dual \widehat G. For every admissible linear form \psi
of the Lie algebra g of G, we can construct an irreducible unitary representation \pi \psi by holomorphic
induction and according to Lipsman (see [9]), every irreducible representation of G arises in this
manner. Then we get a map from the set g\ddagger of the admissible linear forms onto the dual space \widehat G of
G. Note that \pi \psi is equivalent to \pi \psi \prime if and only if \psi and \psi
\prime
are on the same G-orbit, finally we
obtain a bijection between the space g\ddagger /G of admissible coadjoint orbits and the unitary dual \widehat G.
Definition 1. Let G be a (real) Lie group, g its Lie algebra and
\mathrm{e}\mathrm{x}\mathrm{p} : g - \rightarrow G
its exponential map. We say that G is exponential if \mathrm{e}\mathrm{x}\mathrm{p}(g) = G.
c\bigcirc A. MESSAOUD, A. RAHALI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7 945
946 A. MESSAOUD, A. RAHALI
Now, we give our main result in this paper, which is another proof for the continuity of the orbit
mapping (see [11]):
Theorem 1. We assume that G is exponential. Then the orbit mapping
\Theta : g\ddagger /G - \rightarrow \widehat G
is continuous.
This paper is organized as follows. Section 2 is devoted to the description of the unitary dual \widehat G
of G. Section 3 deals with the space of admissible coadjoint orbits g\ddagger /G of G. Theorem 1 is proved
in Section 4.
2. Dual spaces of semidirect product. Throughout this paper, K will denote a connected
compact Lie group acting by automorphisms on a finite dimensional vector space (V, \langle , \rangle ). We write
k.v and A.v (resp., k.\ell and A.\ell ) for the result of applying elements k \in K and A \in k := \mathrm{L}\mathrm{i}\mathrm{e} (K)
to v \in V (resp., to \ell \in V \ast ).
Now, one can form the semidirect product G := K \ltimes V which so-called generalized motion
groups. As a set G = K \times V and the multiplication in this group is given by
(k, v)(h, u) = (kh, v + k.u) \forall (k, v), (h, u) \in G.
The Lie algebra of G is g = k\oplus V (as a vector space) and the Lie algebra structure is given by the
bracket
[(A, a), (B, b)] = ([A,B], A.b - B.a) \forall (A, a), (B, b) \in g.
Under the identification of the dual g\ast of g with k\ast \oplus V \ast , we can express the duality between g and
g\ast as F (A, a) = f(A) + \ell (a) for all F = (f, \ell ) \in g\ast and (A, a) \in g. The adjoint representation
\mathrm{A}\mathrm{d}G and coadjoint representation \mathrm{A}\mathrm{d}\ast G of G are given, respectively, by the following relations:
\mathrm{A}\mathrm{d}G(k, v)(A, a) = (\mathrm{A}\mathrm{d}K(k)A, k.a - \mathrm{A}\mathrm{d}K(k)A.v) \forall (k, v) \in G, (A, a) \in g,
\mathrm{A}\mathrm{d}\ast G(k, v)(f, \ell ) = (\mathrm{A}\mathrm{d}\ast K(k)f + k.\ell \odot v, k.\ell ) \forall (k, v) \in G, (f, \ell ) \in g\ast ,
where \ell \odot v is the element of k\ast defined by
\ell \odot v(A) = \ell (A.v) = - (A.\ell )(v) \forall A \in k, \ell \in V \ast , v \in V.
Note that the map \odot : V \ast \times V - \rightarrow k\ast defined by (\ell \odot v)(A) = \ell (A.v), v \in V, A \in k satisfies a
fundamental equivariance property
\mathrm{A}\mathrm{d}\ast K(k)(\ell \odot v) = (k.\ell )\odot (k.v), k \in K.
Therefore, the coadjoint orbit of G passing through (f, \ell ) \in g\ast is given by
\scrO G
(f,\ell ) =
\Bigl\{ \Bigl(
Ad\ast K(k)f + k.\ell \odot v, k.\ell
\Bigr)
, k \in K, v \in V
\Bigr\}
.
For \ell \in V \ast , we define K\ell := \{ k \in K; k.\ell = \ell \} the isotropy subgroup of \ell in K and the Lie algebra
of K\ell is given by the vector space k\ell = \{ A \in k; A.\ell = 0\} . Let \imath \ell : k\ell \lhook \rightarrow k be the injection map,
then \imath \ast \ell : k\ast - \rightarrow k\ast \ell is the projection map and we have
k\circ \ell = \mathrm{K}\mathrm{e}\mathrm{r} (\imath \ast \ell ), (1)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ANOTHER PROOF FOR THE CONTINUITY OF THE LIPSMAN MAPPING 947
where k\circ \ell is the annihilator of k\ell . If we define the linear map h\ell : k - \rightarrow V \ast by
h\ell (A) := - A.\ell \forall A \in k,
then we have k\ell = \mathrm{K}\mathrm{e}\mathrm{r}(h\ell ). The dual h\ast \ell : V - \rightarrow k\ast of h\ell is given by the relation h\ast \ell (v)(A) =
= h\ell (A)(v) = - (A.\ell )(v), and so h\ast \ell (v) = \ell \odot v \forall \ell \in V \ast , \forall v \in V (for more details see [3]).
The following is a useful lemma from [3], giving a characterization of the annihilator k\circ \ell in terms
of the linear map h\ell .
Lemma 1. Using the previous notations, then we have the equality
k\circ \ell = \mathrm{I}\mathrm{m}(h\ast \ell ).
Here we recall briefly the description of the unitary dual of G via Mackey’s little group theory
(see [10]). For every non-zero linear form \ell on V, we denote by \chi \ell the unitary character of the
vector Lie group V given by \chi \ell = ei\ell . Let \rho be an irreducible unitary representation of K\ell on some
Hilbert space \scrH \rho . The map
\rho \otimes \chi \ell : (k, v) \mapsto - \rightarrow ei\ell (v)\rho (k)
is a representation of the Lie group K\ell \ltimes V such that one induce up so as to get a unitary represen-
tation of G. We denote by \scrH (\rho ,\ell ) := L2(K,\scrH \rho )
\rho the subspace of L2(K,\scrH \rho ) consisting of all the
maps \xi which satisfy the covariance condition
\xi (kh) = \rho (h - 1)\xi (k) \forall k \in K, h \in K\ell .
The induced representation
\pi (\rho ,\ell ) := \mathrm{I}\mathrm{n}\mathrm{d}K\ltimes V
K\ell \ltimes V (\rho \otimes \chi \ell )
is defined on \scrH (\rho ,\ell ) by
\pi (\rho ,\ell )(k, v)\xi (h) = ei\ell (h
- 1.v)\xi (k - 1h),
where (k, v) \in G, h \in K and \xi \in \scrH (\rho ,\ell ). By Mackey’s theory we can say that the induced repre-
sentation \pi (\rho ,\ell ) is irreducible and every infinite dimensional irreducible unitary representation of G
is equivalent to one of \pi (\rho ,\ell ). Moreover, tow representations \pi (\rho ,\ell ) and \pi (\rho \prime ,\ell \prime ) are equivalent if and
only if \ell and \ell
\prime
are contained in the same K -orbit and the representation \rho and \rho
\prime
are equivalent
under the identification of the conjugate subgroups K\ell and K\ell \prime . All irreducible representations of
G which are not trivial on the normal subgroup V, are obtained by this manner. On the other hand,
we denote also by \tau the extension of every unitary irreducible representation \tau of K on G, which
simply defined by \tau (k, v) := \tau (k) for k \in K and v \in V. Let \Omega be a K -orbit in V \ast . We fix \ell \in \Omega
and we define the subset \widehat G(\Omega ) of \widehat G by
\widehat G(\Omega ) = \Bigl\{
\mathrm{I}\mathrm{n}\mathrm{d}K\ltimes V
K\ell \ltimes V (\rho \otimes \chi \ell ); \rho \in \widehat K\ell
\Bigr\}
.
Then we conclude that \widehat G = \widehat K\bigcup \Bigl( \bigcup
\Omega \in \Lambda
\widehat G(\Omega )\Bigr) ,
where \Lambda is the set of the nontrivial orbits in V \ast /K.
In the remainder of this paper, we shall assume that G is exponential, i.e., K\ell is connected for
all \ell \in V \ast . Let \rho \mu be an irreducible representation of K\ell with highest weight \mu . For simplicity, we
shall write \pi (\mu ,\ell ) instead of \pi (\rho \mu ,\ell ) and \scrH (\mu ,\ell ) instead of \scrH (\rho \mu ,\ell ).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
948 A. MESSAOUD, A. RAHALI
We close this section by presenting two results which are being used in the description of the
dual topology of G. These are required for our proof of Theorem 1.
Let N be an Abelian group, and assume that the compact Lie group K acts on the left on N
by automorphisms. As sets, the semidirect product K \ltimes N is the Cartesian product K \times N and the
group multiplication is given by
(k1, x1) \cdot (k2, x2) = (k1k2, x1 + k1x2).
Let \chi be a unitary character of N, and let K\chi be the stabilizer of \chi under the action of K on \widehat N
defined by
(k \cdot \chi )(x) = \chi (k - 1x).
If \rho is an element of \widehat K\chi , then the triple (\chi , (K\chi , \rho )) is called a cataloguing triple. From the
notations of [2], we denote by \pi (\chi ,K\chi , \rho ) the induced representation \mathrm{I}\mathrm{n}\mathrm{d}K\ltimes N
K\chi \ltimes N (\rho \otimes \chi ). Referring
to [2, p. 187], we have the following proposition.
Proposition 1. The mapping (\chi , (K\chi , \rho )) - \rightarrow \pi (\chi ,K\chi , \rho ) is onto \widehat K \ltimes N.
We denote by \scrA (K) the set of all pairs (K \prime , \rho \prime ), where K \prime is a closed subgroup of K and \rho \prime
is an irreducible representation of K \prime . We equip \scrA (K) with the Fell topology (see [6]). Therefore,
every element in \widehat K \ltimes N can be catalogued by elements in the topological space \widehat N \times \scrA (K). Larry
Baggett has given an abstract description of the topology of the dual space of a semidirect product
of a compact group with an Abelian group in terms of the Mackey parameters of the dual space (see
[2], Theorem 6.2-A). The following result provides a precise and neat description of the topology of
\widehat K \ltimes N.
Theorem 2. Let Y be a subset of \widehat K \ltimes N and \pi an element of \widehat K \ltimes N. Then \pi is weakly
contained in Y if and only if there exist: a cataloguing triple (\chi , (K\chi , \rho )) for \pi , an element (K \prime , \rho \prime )
of \scrA (K), and a net \{ (\chi n, (K\chi n , \rho n))\} of cataloguing triples such that:
(i) for each n, the irreducible unitary representation \pi (\chi n,K\chi n , \rho n) of K \ltimes N is an element
of Y ;
(ii) the net \{ (\chi n, (K\chi n , \rho n))\} converges to (\chi , (K \prime , \rho \prime ));
(iii) K\chi contains K \prime , and the induced representation \mathrm{I}\mathrm{n}\mathrm{d}
K\chi
K\prime (\rho \prime ) contains \rho .
3. Admissible coadjoint orbits of semidirect product. We keep the notations of Section 2. Fix
a non-zero linear form \ell \in V \ast , and we consider an irreducible representation \rho \mu of K\ell with highest
weight \mu . Then the stabilizer G\psi of \psi = (\mu , \ell ) in G is given by
G\psi =
\Bigl\{
(k, v) \in G; (\mathrm{A}\mathrm{d}\ast K(k)\mu + k.\ell \odot v, k.\ell ) = (\mu , \ell )
\Bigr\}
=
=
\Bigl\{
(k, v) \in G; k \in K\ell ,\mathrm{A}\mathrm{d}
\ast
K(k)\mu + \ell \odot v = \mu
\Bigr\}
=
=
\Bigl\{
(k, v) \in G; k \in K\ell , \mathrm{A}\mathrm{d}
\ast
K(k)\mu = \mu
\Bigr\}
since \imath \ast \ell (\ell \odot v) = 0 (see Lemma 1 and the equality (1)). Thus, we have G\psi = K\psi \ltimes V\psi , then \psi is
aligned (see [9]). A linear form \psi \in g\ast is called admissible if there exists a unitary character \chi of the
identity component of G\psi such that d\chi = i\psi | g\psi . According to Lipsman (see [9]), the representation
of G obtained by holomorphic induction from (\mu , \ell ) is equivalent to the representation \pi (\mu ,\ell ). Let \tau \lambda
be an irreducible representation of K with highest weight \lambda , then the representation of G obtained by
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ANOTHER PROOF FOR THE CONTINUITY OF THE LIPSMAN MAPPING 949
holomorphic induction from (\lambda , 0) is equivalent to \tau \lambda . The coadjoint orbit of G through (\lambda , 0) \in g\ast
is denoted by \scrO G
\lambda . It is clear that \scrO G
\lambda is an admissible coadjoint orbit of G. We denote by g\ddagger \subset g\ast
the set of all admissible linear forms on g. The quotient space g\ddagger /G is called the space of admissible
coadjoint orbits of G. Moreover, one can check that g\ddagger /G is the union of the set of all orbits \scrO G
(\mu ,\ell )
and the set of all orbits \scrO G
\lambda .
We conclude this section by recalling needed results. Let L be a closed subgroup of K. By
TK and TL be maximal tori, respectively, in K and L such that TL \subset TK . Their corresponding
Lie algebras are denoted by tk and tl. We denote by WK and WL the Weyl groups of K and
L associated, respectively, to the tori TK and TL. Notice that every element \lambda \in PK takes pure
imaginary values on tk, where PK is the integral weight lattice of TK . Hence such an element
\lambda \in PK can be considered as an element of (itk)
\ast . Let C+
K be a positive Weyl chamber in (itk)
\ast ,
and we define the set P+
K of dominant integral weights of TK by P+
K := PK \cap C+
K . For \lambda \in P+
K ,
denote by \scrO K
\lambda the K -coadjoint orbit passing through the vector - i\lambda . It was proved by Kostant in
[7], that the projection of \scrO K
\lambda on t\ast k is a convex polytope with vertices - i(w.\lambda ) for w \in WK , and
that is the convex hull of - i(WK .\lambda ). For the same manner, we fix a positive Weyl chamber C+
L in
t\ast l and we define the set P+
L of dominant integral weights of TL.
Also we denote by \imath \ast l the \BbbC -linear extension of both the natural projection of k\ast onto l\ast and the
natural projection of t\ast k onto t\ast l . Consider tow irreducible representations \tau \lambda \in \widehat K and \rho \mu \in \widehat L with
respective highest weights \lambda \in P+
K and \mu \in P+
L . We have the following result.
Lemma 2. If \mu = i\ast l (s.\lambda ) with s \in WK , then \tau \lambda occurs in the induced representation \mathrm{I}\mathrm{n}\mathrm{d}KL (\rho \mu ).
We refer to [1], for the proof of this lemma.
4. Main results. We shall freely use the notations of the previous sections.
Remark 1. We have the following convergence:
\ell m - \rightarrow \ell ,
K\ell m \subseteq K\ell .
To study the convergence in the quotient space g\ddagger /G, we need to the following result (see [8,
p. 135] for the proof).
Lemma 3. Let G be a unimodular Lie group with Lie algebra g and let g\ast be the vector dual
space of g. We denote g\ast /G the space of coadjoint orbits and by pG : g\ast - \rightarrow g\ast /G the canonical
projection. We equip this space with the quotient topology, i.e., a subset V in g\ast /G is open if
and only if p - 1
G
(V ) is open in g\ast . Therefore, a sequence (\scrO G
n )n of elements in g\ast /G converges
to the orbit \scrO G in g\ast /G if and only if for any l \in \scrO G, there exist ln \in \scrO G
n , n \in \BbbN , such that
l = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow +\infty ln.
Now, we are in position to prove the following propositions.
Proposition 2. Let
\bigl(
\scrO G
(\mu m,\ell m)
\bigr)
m
be a sequence in g\ddagger /G. If
\bigl(
\scrO G
(\mu m,\ell m)
\bigr)
m
converges to \scrO G
(\mu ,\ell )
in g\ddagger /G, then we have: (\ell m)m converges to \ell and for m large enough, \rho \mu \in \mathrm{I}\mathrm{n}\mathrm{d}K\ell K\ell m
(\rho \mu m).
Proof. We assume that the sequence of admissible coadjoint orbits
\bigl(
\scrO G
(\mu m,\ell m)
\bigr)
m
converges to
\scrO G
(\mu ,\ell ) in g\ddagger /G. By referring to [3], we show that the coadjoint orbit \scrO G
(\mu ,\ell ) is always obtained
by symplectic induction from the coadjoint orbit M = \scrO H
(\mu ,\ell ) of H := K\ell \ltimes V passing through
(\mu , \ell ) \in k\ast \ell \oplus V \ast (k\ell \ltimes V := \mathrm{L}\mathrm{i}\mathrm{e}(H)), i.e.,
\scrO G
(\mu ,\ell ) =Mind := J - 1\widetilde M (0)/H, (2)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
950 A. MESSAOUD, A. RAHALI
where J\widetilde M : \widetilde M =M \times T \ast G - \rightarrow k\ast \ell \ltimes V \ast is the momentum map of \widetilde M and the zero level set J - 1\widetilde M (0)
is given by
J - 1\widetilde M (0) =
\Bigl\{ \Bigl(
(\mathrm{A}\mathrm{d}\ast K(k)\mu , \ell ), g, (\mathrm{A}\mathrm{d}\ast K(k)\mu + \ell \odot v, \ell )
\Bigr)
, k \in K\ell , g \in G, v \in V
\Bigr\}
.
Let \varphi M be the action of H on M, hence H acts on \widetilde M =M \times T \ast G by \varphi \widetilde M as follows:
\varphi \widetilde M (h)(\alpha , g, f) =
\bigl(
\varphi M (h)(\alpha ), gh - 1,\mathrm{A}\mathrm{d}\ast H(h)f
\bigr)
(3)
for all h \in H, (\alpha , g, f) \in M \times T \ast G. By identifying g\ast with the left-invariant 1-form on G. Then
we can write T \ast G \sim = G\times g\ast .
Using Lemma 3 and by combining (2) with (3), then there exist sequences km, hm \in K\ell m ,
vm, wm \in V, and gm \in G such that the sequence (\phi m)m defined by
\phi m = \varphi \widetilde M (km, vm)
\Bigl(
(\mathrm{A}\mathrm{d}\ast K(hm)\mu
m, \ell m), gm, (\mathrm{A}\mathrm{d}
\ast
K(hm)\mu
m + \ell m \odot wm, \ell m)
\Bigr)
=
=
\Bigl(
\mathrm{A}\mathrm{d}\ast K(kmhm)\mu
m + \imath \ast \ell m(\ell m \odot vm), \ell m
\bigr)
, gm(km, vm)
- 1,
(\mathrm{A}\mathrm{d}\ast K(kmhm)\mu
m +\mathrm{A}\mathrm{d}\ast K(km)(\ell m \odot wm) + \ell m \odot vm, \ell m)
\Bigr)
converges to
\bigl(
(\mu , \ell ), eG, (\mu , \ell )
\bigr)
. It follows that
\ell m - \rightarrow \ell
and
\mathrm{A}\mathrm{d}\ast K(kmhm)\mu
m + \imath \ast \ell m(\ell m \odot vm) - \rightarrow \mu (4)
as n - \rightarrow +\infty . By compactness of K we may assume that (kmhm)m converges to p \in K\ell n \subset K\ell .
By using the fact that \imath \ast \ell m(\ell m \odot vm) = 0, we, from (4), obtain that
\mu m = Ad\ast (p - 1)\mu
for m large enough. Furthermore, we known that there exists an element s \in WK\ell such that
\mathrm{A}\mathrm{d}\ast (p - 1)\mu = s.\mu . Hence \mu m = s.\mu for m large enough and we conclude by Lemma 2 that for m
large enough, \rho \mu \in \mathrm{I}\mathrm{n}\mathrm{d}K\ell K\ell m
(\rho \mu m).
Proposition 2 is proved.
Proposition 3. If the sequence
\bigl(
\scrO G
(\mu m,\ell m)
\bigr)
m
converges to \scrO G
\lambda in g\ddagger /G, then we have: (\ell m)m
converges to 0 and for m large enough, \tau \lambda \in \mathrm{I}\mathrm{n}\mathrm{d}KK\ell m (\rho \mu
m).
Proof. We use the notations and proceedings of the proof of the last proposition. Let us as-
sume that the sequence
\bigl(
\scrO G
(\mu m,\ell m)
\bigr)
m
converges to \scrO G
\lambda . Then there exist sequences km, hm \in K\ell m ,
vm, wm \in V, and gm \in G such that the sequence (\Psi m)m defined by
\Psi m = \varphi \widetilde M (km, vm)
\Bigl(
(\mathrm{A}\mathrm{d}\ast K(hm)\mu
m, \ell m), gm, (\mathrm{A}\mathrm{d}
\ast
K(hm)\mu
m + \ell m \odot wm, \ell m)
\Bigr)
=
=
\Bigl(
\mathrm{A}\mathrm{d}\ast K(kmhm)\mu
m + \imath \ast \ell m(\ell m \odot vm), \ell m
\bigr)
, gm(km, vm)
- 1,
(\mathrm{A}\mathrm{d}\ast K(kmhm)\mu
m +\mathrm{A}\mathrm{d}\ast K(km)(\ell m \odot wm) + \ell m \odot vm, \ell m)
\Bigr)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ANOTHER PROOF FOR THE CONTINUITY OF THE LIPSMAN MAPPING 951
converges to
\bigl(
(\lambda , 0), eG, (\lambda , 0)
\bigr)
. From the above facts, we conclude the following convergence:
\ell m - \rightarrow 0, (5)
\mathrm{A}\mathrm{d}\ast (kmhm)\mu
m - \rightarrow \lambda . (6)
By assumption that the sequence (kmhm)m converges to p \in K\ell m , we obtain, from (6), that \mu m =
= \mathrm{A}\mathrm{d}\ast (p - 1)\lambda for m large enough. Hence there exists w \in WK , such that \mu m = w.\lambda for m large
enough. Lemma 2 allows us to derive that \tau \lambda \in \mathrm{I}\mathrm{n}\mathrm{d}KK\ell m (\rho \mu
m) for large m.
Proposition 3 is proved.
Proposition 4. If (\scrO G
\lambda m)m converges to \scrO G
\lambda in g\ddagger /G, then \lambda m = \lambda for large m.
Proof. Suppose that (\scrO G
\lambda m)m converges to \scrO G
\lambda in g\ddagger /G, then there exists (km)m \subset K such that
\mathrm{A}\mathrm{d}\ast K(km)\lambda
m - \rightarrow \lambda as m - \rightarrow +\infty .
By compactness of K we may assume that (km)m converges to k \in K. Then we obtain \lambda m =
= \mathrm{A}\mathrm{d}\ast K(k - 1)\lambda for m large enough. Hence there exists w \in WK such that \mathrm{A}\mathrm{d}\ast K(k - 1) = w.\lambda for
m large enough. It follows that \lambda m = w.\lambda for m large enough. Since the weights \lambda m and \lambda are
contained in the set iC+
K and since each WK -orbit in k\ast intersects the closure iC+
K in exactly one
point, it follows that \lambda m = \lambda for m large enough.
Proposition 4 is proved.
Combining the above Propositions 2, 3 and 4 with Baggett’s theorem (Theorem 2), we obtain our
result (Theorem 1).
References
1. D. Arnal, M. Ben Ammar, M. Selmi, Le problème de la réduction à un sous-groupe dans la quantification par
déformation, Ann. Fac. Sci. Toulouse, 12, 7 – 27 (1991).
2. W. Baggett, A description of the topology on the dual spaces of certain locally compact groups, Trans. Amer. Math.
Soc., 132, 175 – 215 (1968).
3. P. Baguis, Semidirect product and the Pukanszky condition, J. Geom. and Phys., 25, 245 – 270 (1998).
4. M. Ben Halima, A. Rahali, On the dual topology of a class of Cartan motion groups, J. Lie Theory, 22, 491 – 503
(2012).
5. M. Elloumi, J. Ludwig, Dual topology of the motion groups SO(n)\ltimes \BbbR n , Forum Math., 22, 397 – 410 (2008).
6. J. M. G. Fell, Weak containment and induced representations of groups (II), Trans. Amer. Math. Soc., 110, 424 – 447
(1964).
7. B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Supér, 6, 413 – 455
(1973).
8. H. Leptin, J. Ludwig, Unitary representation theory of exponential Lie groups, de Gruyter, Berlin (1994).
9. R. L. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures et
Appl., 59, 337 – 374 (1980).
10. A. Rahali, Dual topology of generalized motion groups, Math. Rep., 20(70), 233 – 243 (2018).
11. A. Messaoud, A. Rahali, On the continuity of the Lipsman mapping of semidirect products, Rev. Roum. Math. Pures
et Appl., 3(63), 249 – 258 (2018).
Received 10.05.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
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| id | umjimathkievua-article-548 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:03Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f9/880cc59181638fcea2cb79fb21e92cf9.pdf |
| spelling | umjimathkievua-article-5482022-03-26T11:01:52Z Another proof for the continuity of the Lipsman mapping Another proof for the continuity of the Lipsman mapping Messaoud, A. Rahali, A. Messaoud, A. Rahali, A. Lie groups semidirect product unitary representations coadjoint orbits symplectic induction Lie groups semidirect product unitary representations coadjoint orbits symplectic induction UDC 515.1 We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle , \rangle$. By $\hat G $ we denote the unitary dual of $G$ and by ${\mathfrak{g}^{ \ddagger} /} G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G$. It was pointed out by Lipsman that the correspondence between $\hat{G} $ and ${\mathfrak{g}^{ \ddagger} /} G$ is bijective. Under some assumption on $G$, we give another proof for the continuity of the orbit mapping (Lipsman mapping)$$\Theta : {\mathfrak{g}^{ \ddagger} /} G - \rightarrow \hat{G} .$$ УДК 515.1 Ще одне доведення неперервностi вiдображення Лiпсмана Розглядається напiвпрямий добуток $G = K \ltimes V$, де $K$ є зв’язною компактною групою Лi автоморфiзмiв, що дiють на скiнченновимiрному дiйсному векторному просторi $ V$ iз внутрiшнiм добутком $\langle ,\rangl$$\Theta : {\frak{g}^{ \ddagger} /} G - \rightarrow \widehat{G} .$$ Institute of Mathematics, NAS of Ukraine 2020-07-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/548 10.37863/umzh.v72i7.548 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 7 (2020); 945-951 Український математичний журнал; Том 72 № 7 (2020); 945-951 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/548/8730 |
| spellingShingle | Messaoud, A. Rahali, A. Messaoud, A. Rahali, A. Another proof for the continuity of the Lipsman mapping |
| title | Another proof for the continuity of the Lipsman mapping |
| title_alt | Another proof for the continuity of the Lipsman mapping |
| title_full | Another proof for the continuity of the Lipsman mapping |
| title_fullStr | Another proof for the continuity of the Lipsman mapping |
| title_full_unstemmed | Another proof for the continuity of the Lipsman mapping |
| title_short | Another proof for the continuity of the Lipsman mapping |
| title_sort | another proof for the continuity of the lipsman mapping |
| topic_facet | Lie groups semidirect product unitary representations coadjoint orbits symplectic induction Lie groups semidirect product unitary representations coadjoint orbits symplectic induction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/548 |
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