  For a complex connected reductive group G, we classify the simple modules
whose cone of primitive vectors admits a nontrivial G-invariant deformation. We
relate this classification to that of simple Jordan algebras, and to that (due
to Akhiezer) of smooth projective varieties whose orbits under the action of a
connected affine algebraic group are a divisor and its complementary. Our main
tool is the invariant Hilbert scheme of Alexeev-Brion; we determine the first
examples of it. We also determine the infinitesimal deformations (non
necessarily G-invariant) of the cones of primitive vectors; they turn out to be
trivial for most simple modules.

Algebraic Geometry;Representation Theory