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

Let G be a compact subgroup of
G
L
n
(
R
)
acting linearly on a finite dimensional 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 ([O]).