For a connected reductive group $G$, and a finite dimensional $G$-module $V$, Alexeev and M. Brion have built the invariant Hilbert scheme: it parametrizes $G$-stable closed subschemes of $V$ affording a fixed, multiplicity-finite representation of $G$ in their coordinate ring. We shall describe this scheme in the simplest case, where it parametrizes invariant deformations of the cone of primitive vectors of a simple $G$-module. The classification we get is related to those of simple Jordan algebras and of wonderful varieties of rank one whose open orbit is affine.