Differential operators on the supercircle S1|2 and symbol map

We consider the supercircle [Formula: see text] equipped with the standard contact structure. The conformal Lie superalgebra [Formula: see text] acts on [Formula: see text] as the Lie superalgebra of contact vector fields; it contains the M[Formula: see text]bius superalgebra [Formula: see text]. We study the space of linear differential operators on weighted densities as a module over [Formula: see text]. We introduce the canonical isomorphism between this space and the corresponding space of symbols. This result allows us to give, in contrast to the classical setting, a classification of the [Formula: see text]-modules [Formula: see text] of linear differential operators of order [Formula: see text] acting on the superspaces of weighted densities. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of differential operators on the real line, Funct. Anal. Appl. 35(1) (2001) 13–18.]

Bases for modules of differential operators

International audience It is well-known that the derivation modules of Coxeter arrangements are free. Holm began to study the freeness of modules of differential operators on hyperplane arrangements. In this paper, we study the cases of the Coxter arrangements of type A, B and D. In this case, we prove that the modules of differential operators of order 2 are free. We give examples of all the 3-dimensional classical Coxeter arrangements. Two keys for the proof are ``Cauchy–Sylvester's theorem on compound determinants'' and ``Saito–Holm's criterion''. Il est connu que les modules de la dérivation d'arrangements de Coxeter sont libres. Holm a commencè à étudier les modules libres des opérateurs différentiels sur des compositions d'hyperplans. Dans cet article, nous étudions les cas des compositions de Coxter les types A, B et D. Dans ce cas, nous prouvons que les modules d’opérateurs différentiels d'ordre 2 sont libres. Nous donnons des exemples de toutes les compositions de Coxeter classiques de dimension 3. Les deux points clefs pour la preuve sont le théorème de Cauchy–Sylvester sur déterminants composés et le critère de Saito–Holm.