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

Abstract

Let $G$ be a compact subgroup of $G{L}_n(\mathbb{R})$ acting linearly on a finite dimensional vector space $E$. B. Malgrange has shown that the space ${\mathcal{C}}^{\infty }({\mathbb{R}}^n,E{)}^G$ of ${\mathcal{C}}^{\infty }$ and $G$-covariant functions is a finite module over the ring ${\mathcal{C}}^{\infty }({\mathbb{R}}^n{)}^G$ of ${\mathcal{C}}^{\infty }$ and $G$-invariant functions. First, we generalize this result for the Schwartz space $\mathcal{S}({\mathbb{R}}^n,E{)}^G$ 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]).