Jets de fonctions differentiables sur le dual d'un groupe de Lie nilpotent. (Jets of differentiable functions on the dual of a nilpotent Lie group)
Let G be a simply connected nilpotent Lie group. In the setting of Alain Connes' non-commutative differential geometry, it is natural to take the convolution algebra \({\mathcal S}={\mathcal S}(G)\subset L^ 1(G)\) of Schwartz functions on G as the algebra of differentiable functions on the dual G{\^{\ }}. In this article, we define the notion of a ''Taylor series'' of such a function at a point (E,\(\pi)\) of G{\^{\ }}. More precisely, let M be the annihilator of E in \({\mathcal S}\). We show that the powers of M are closed ideals of \({\mathcal S}\), and we define \({\mathcal S}/M^{\infty}=\lim_{\leftarrow}{\mathcal S}/M^ n\) to be the algebra of jets of differentiable functions at the point (E,\(\pi)\). We show that \({\mathcal S}/M^{\infty}\) is Morita-equialent, in a suitable sense, to an algebra of ''non-commutative formal power series'' which is a projective limit of finite dimensional algebras over \({\mathbb{C}}\); we use a theorem of Gabriel on the unicity of the ''sober'' algebra attached to a finite abelian category. We give explicitly the formula for the Taylor series expansion of a function \(\phi\in {\mathcal S}\) at the trivial representation.
scientific article; zbMATH DE number 3957910
scientific article