an:00508680
Zbl 0798.22004
Ludwig, Jean; Zahir, Hamid
On the nilpotent \(*\)-Fourier transform
EN
Lett. Math. Phys. 30, No. 1, 23-34 (1994).
00017318
1994
j
22E27 43A30
simply connected nilpotent Lie group; Fourier transform; Schwartz space; diffeomorphic parametrization; Lie algebra; distributions; convolution product; Moyal \(*\)-product
For a connected and simply connected nilpotent Lie group \(G\), one considers the Fourier transform \({\mathcal E}: {\mathcal S}(G) \to C^ \infty(V,{\mathcal S}(\mathbb{R}^{2d}))\):
\[
{\mathcal E}(f)(\lambda,p,q) = \int_ G e^{-ia(g,\lambda,p,q)} f(g)dg,
\]
where \({\mathcal S}(G)\) is the Schwartz space of \(G\), \(V \times {\mathbb{R}}^{2d} \to {\mathcal O}\) an adapted diffeomorphic parametrization for a \(G\)-invariant Zariski open set \(\mathcal O\) on the dual space \(g^*\) of the Lie algebra \(g = \text{Lie}(G)\) and \(a(.,.,.,.)\) a real function, polynomial in \(p\) and \(q\) and rational in \(\lambda\), with singularities outside \(V\). The authors prove the surjectivity of this transformation (Corollary 2.2.4) and therefore extend this Fourier transform to distributions. This leads them in particular to have some interesting results, for example \({\mathcal E}(\delta_ x) = e^{ia(.,.,.,.)}\), it transforms the convolution product on group \(G\) to the Moyal \(*\)-product on \(V \times \mathbb{R}^{2d}\), \({\mathcal E}(T * f) = {\mathcal E}(T) * {\mathcal E}(f)\).
Diep Do Ngoc (Hanoi)