One-dimensional infinitesimal-birational duality through differential operators

Abstract

Abstract. The structure of filtered algebras of Grothendieck's differential operators on a smooth fat point in a curve and graded Poisson algebras of their principal symbols is explicitly determined. A related infinitesimal-birational duality realized by a Springer type resolution of singularities and the Fourier transformation is presented. This algebrogeometrical duality is quantized in appropriate sense and its quantum origin is explained.It seems that our ship is arriving at a coast, but the land is still hidden in the fog.