On covariant functions and distributions under the action of a compact group

Let G be a compact subgroup of GL n ( R ) acting linearly on a finite dimensional complex vector space E. B. Malgrange has shown that the space C ∞ ( R n , E ) G of C ∞ and G-covariant functions is a finite module over the ring C ∞ ( R n ) G of C ∞ and G-invariant functions. First, we generalize this result for the Schwartz space S ( R n , E ) G of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in [4].