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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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