  Given an affine scheme X with an action of a reductive group G and a
G-linearized coherent sheaf M, we construct the ``invariant Quot scheme'' that
parametrizes the quotients of M whose space of global sections is a direct sum
of simple G-modules with fixed finite multiplicities.
  Then we determine the invariant Quot scheme in a simple situation, where X is
the cone of primitive vectors of a simple G-module and M is the free sheaf on X
generated by another simple G-module. This invariant Quot scheme has only one
point, that is reduced in most of the cases. The only cases where it is not
reduced occur when X is the cone of primitive vectors of a quadratic vector
space V of odd dimension, under the action of Spin(V).
