## On the nilpotent $$*$$-Fourier transform.(English)Zbl 0798.22004

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 $$V \times \mathbb{R}^{2d}$$, $${\mathcal E}(T * f) = {\mathcal E}(T) * {\mathcal E}(f)$$.

### MSC:

 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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### References:

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