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Adapted Fourier transform of Schwartz spaces for certain nilpotent Lie groups. (English) Zbl 0714.43005
For the Schwartz class functions on a symplectic manifold one can define a star product *. In particular for a connected and simply connected nilpotent Lie group G with Lie algebra $${\mathfrak g}$$, we can consider a smooth star product * of the well-behaved functions on some Zariski open invariant set $$F=\coprod_{\lambda \in V}{\mathfrak O}^{\lambda}$$ of coadjoint orbits on the dual to $${\mathfrak g}$$ vector space $${\mathfrak g}^*$$. With this *-product on each coadjoint orbit $${\mathfrak O}^{\lambda}\cong {\mathbb{R}}^{2k}_{(p,q)}$$ one can introduce the weight $$\Omega =2^{(1/2)\dim {\mathcal O}^{\lambda}}e^{-(p^ 2+q^ 2)}$$, the weighted space $$L^ 2({\mathfrak O}^{\lambda})*\Omega$$, the left *-product representation $$T^{\lambda}$$ of the Schwartz class function algebra $${\mathcal S}({\mathfrak O}^{\lambda})$$ and finally, the intertwining operator between $$L^ 2({\mathcal O}^{\lambda})*\Omega$$ and the left regular representation space $$L^ 2_{polar}({\mathcal O}^{\lambda})\simeq L^ 2({\mathbb{R}}^ k)$$. Then the nilpotent (adapted) Fourier transform of a Schwartz class function $$f\in {\mathcal S}(G)$$ is defined by $\Theta (f)|_{{\mathcal O}^{\lambda}}:=(T^{\lambda})^{-1}\circ \pi^{\lambda}(f),$ where $$\pi^{\lambda}$$ is the irreducible representation of G on $$L^ 2_{polar}({\mathcal O}^{\lambda})$$, corresponding to the orbit $${\mathcal O}^{\lambda}$$. This Fourier transform can be represented by some oscillating integral formula.
In a previous work [C. R. Acad. Sci., Paris, Sér. I 305, 769-772 (1987; Zbl 0643.43003)] the author has proved that $$\Theta$$ : $${\mathcal S}(G)\to C^{\infty}(V,{\mathcal S}({\mathbb{R}}^{2k}_{(p,q)}))$$ is a continuous map from $${\mathcal S}(G)$$ into the space of $$C^{\infty}$$-functions on V with values in the Schwartz space on the standard symplectic space $$R^{2k}$$ with coordinates $$(p_ 1,...,p_ k,q^ 1,...,q^ k)=(p,q).$$
In the paper under review the author gives some condition (H) guaranting the density of $$\Theta$$ ($${\mathcal S}(G))$$ in $$C^{\infty}(V,{\mathcal S}({\mathbb{R}}^{2k}_{(p,q)}))$$ (Theorem 5). It must be remarked some related results of Nghiem Xuan Hai based on the Generalized Gel’fand-Kirillov conjecture and the Schrödinger representation [C. R. Acad. Sci., Paris, Sér. I 293, 381-384 (1981; Zbl 0472.43008)].
Reviewer: Do Ngoc Diep

##### MSC:
 43A32 Other transforms and operators of Fourier type 43A80 Analysis on other specific Lie groups 22E25 Nilpotent and solvable Lie groups
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##### References:
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