Representations de l'algebre de Lie conforme sur l'espace des densites tensorielles sur la sphere

Abstract

We consider the space of tensor densities on the n-dimensional sphere with degree lambda (or, equivalently, of conformal densities with degree lambda). This space is a module over the group of diffeomorphisms, and consequently over the Lie algebra of vector fields, on the sphere, and we first prove that as a module over the conformal group, it is infinitesimally equivalent to an induced module from the principal series, precisely from the nonunitary spherical series on the space of smooth functions on the sphere. In function of lambda,(g,K)-simple and unitary submodules of the space of tensor densities are therefore classified.