Courants invariants par une action propre

The purpose of this paper is to characterise the invariant sections-distributions by a proper action. More precisely, we show that if G is a connected Lie group acting on a differentiable vector bundle E→V such that the induced action on V is proper, then the topological vector space of the G-invariant linear functionals (on the space of C ^∞ sections with compact support) equipped with the induced weak-topology (resp. the strong-topology), is isomorphic to the weak (resp. strong) topological dual of the space (of all G-invariant sections σ with compact quotient supp(σ)/G) equipped with a suitable topology; this coincides with the usual C ^∞-topology if the orbit space is compact, and with the Schwartz-topology if the group G is compact. Received: 8 June 1998 / Revised version: 22 September 1998