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Fourier transform of \({\mathcal D}_{\chi,\mathbb{Q}}^ † (\infty)\)-modules. (Transformation de Fourier des \({\mathcal D}_{\chi,\mathbb{Q}}^ † (\infty)\)-modules.) (French) Zbl 0872.14011
Let \(\mathcal V\) be a complete discrete valuation ring of unequal characteristics \((0,p)\), \(K\) its fraction field and \(\mathcal X\) the formal projective space over \(\mathcal V\). In this note, the author studies the geometric Fourier transform \({\mathcal F}({\mathcal M})\) of a bounded coherent complex \(\mathcal M\) of \({\mathcal D}^\dagger_{{\mathcal X},\mathbf Q}(\infty)\)-modules. She shows that \({\mathcal F}({\mathcal M})\) coincides with the naive Fourier transform coming from the automorphism \(F\) of the weak completion \(A_N(K)^\dagger\) of the Weyl algebra given by \(F(\partial_{x_i})=\pi x_i, F(x_i) = -\partial_{x_i}/\pi\), where \(\pi^{p-1} = -p\).
See also the reviewer’s remark in the preceding review [C. Huyghe, C. R. Acad. Sci., Paris, Sér. I 321, No. 5, 587-590 (1995)].

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials