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We study Demazure modules which occur in a level $\ell$ irreducibleintegrable representation of an affine Lie algebra. We also assume that theyare stable under the action of the standard maximal parabolic subalgebra of theaffine Lie algebra. We prove that such a module is isomorphic to the fusionproduct of "prime" \ Demazure modules, where the prime factors are indexed bydominant integral weights which are either a multiple of $\ell$ or take valueless than $\ell$ on all simple coroots. Our proof depends on a technical resultwhich we prove in all the classical cases and $G_2$. Calculations withmathematica show that this result is correct for small values of the level.Using our result, we show that there exist generalizations of $Q$--systems topairs of weights where one of the weights is not necessarily rectangular and isof a different level. Our results also allow us to compare the multiplicitiesof an irreducible representation occuring in the tensor product of certianpairs of irreducible representations, i.e., we establish a version of Schurpositvity for such pairs of irreducible modules for a simple Lie algebra.

A Steinberg type decomposition theorem for higher level Demazure modules