
Journal of Lie Theory 32 (2022), No. 3, 879898 Copyright Heldermann Verlag 2022 A DixmierMalliavin Theorem for Lie Groupoids Michael D. Francis Department of Mathematics, University of Western Ontario, Middlesex College, London, Canada mfranc65@uwo.ca A famous theorem of DixmierMalliavin asserts that every smooth, compactlysupported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. The analytical heavy lifting is done by a lemma in the original work of DixmierMalliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from functions vanishing to given order on an invariant submanifold of the unit space. Keywords: DixmierMalliavin, Lie groupoid, convolution. MSC: 22A22, 58H05. [ Fulltextpdf (188 KB)] for subscribers only. 