Abdelmoula, Lobna Baklouti, Ali Bouaziz, Yasmine
Published in
Advances in Pure and Applied Mathematics

Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for Ď in G ^ {\widehat{G}} , the unitary dual of G, sends smooth vectors of the representation space of Ď to đ° â˘ ( đ¤ ) * {{\mathcal{U}(\mathfrak{g})}^{*}} , the dual vector space of đ° â˘ ( đ¤ ) {\mathcal{U}(\mathfrak{g})} . The convex hull of the image of ...

Poguntke, Detlev
Published in
Advances in Pure and Applied Mathematics

It was one of great successes of Kirillov's orbit method to see that the unitary dual of an exponential Lie group is in bijective correspondence with the orbit space associated with the linear dual of the Lie algebra of the group in question. To show that this correspondence is an homeomorphism turned out to be unexpectedly difficult. Only in 1994 ...

Baklouti, Ali Lahyani, Dhoha
Published in
Advances in Pure and Applied Mathematics

So far, the uncertainty principles for solvable non-exponential Lie groups have been treated only in few cases. The first author and Kaniuth produced an analogue of Hardy's theorem for a diamond Lie group, which is a semi-direct product of âd with the Heisenberg group â 2d+1 ${\mathbb {H}_{2d+1}}$ In this setting, we formulate and prove in this pap...

Inoue, Junko
Published in
Advances in Pure and Applied Mathematics

Let G be an exponential solvable Lie group with Lie algebra đ¤ defined by a semi-direct product of â and ân. We study a holomorphically induced representation Ď of G from a real linear form f of đ¤ and a 1-dimensional complex subalgebra đĽ of đ¤â such that the space đĽ + đĽĚ
generates đ¤â. Under some assumptions of structure of the Lie algebra, we obtain ...

Baklouti, Ali Ghaouar, Sonia Khlif, Fatma
Published in
Advances in Pure and Applied Mathematics

Let â2n+1r be the reduced Heisenberg Lie group, G = â2n+1r Ă â2n+1r be the (4n + 2)-dimensional Lie group and ÎG = {(x,x) â G : x â â2n+1r} be the diagonal subgroup of G. Given any discontinuous subgroup Î â G for G/ÎG, we provide a layering of the parameter space â(Î, G, ÎG), which is shown to be endowed with a smooth manifold structure, we also s...

Oussa, Vignon
Published in
Forum Mathematicum

Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra đŤ such that đŤ=đâđâđˇ${\mathfrak {n=a\oplus b\oplus z}}$ , [đ,đ]âđˇ${[ \mathfrak {a},\mathfrak {b}] \subseteq \mathfrak {z}}$ , the algebras đ,đ,đˇ${\mathfrak {a},\mathfrak {b,z}}$ are abelian, đ=â- span {X 1 ,X 2 ,...,X d }${\mathfrak {a}=\mathbb {R}...

Oussa, Vignon
Published in
Advances in Pure and Applied Mathematics

Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie algebra đŤ${\mathfrak {n}}$ is an n-dimensional vector space over the reals. Moreover, đŤ=đˇâđâđ${\mathfrak {n=z}\oplus \mathfrak {b}\oplus \mathfrak {a}}$ , đˇ${\mathfrak {z}}$ is the center of đŤ${\mathfrak {n}}$ , đˇ=âZ n-2d ââZ n-2d-1 ââŻââZ 1 ${\mathfra...

Baklouti, Ali Fujiwara, Hidenori Ludwig, Jean
Published in
Forum Mathematicum

Let G be a real solvable exponential Lie group with Lie algebra đ¤ and let fâđ¤ * ${ f\in \mathfrak {g}^* }$ . We take two polarizations đ j ${ \mathfrak {p}_j}$ , j = 1,2, at f which meet the Pukanszky condition. Let P j :=exp(đ j )${ P_j:=\exp ( \mathfrak {p}_j)}$ , j = 1,2, be the associated subgroups in G. The linear functional f defines unitary ...