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

TL;DR

This paper extends Malgrange's finite module result from smooth to Schwartz covariant functions and shows that covariant distributions can be decomposed into invariant parts multiplied by covariant polynomials, generalizing Oksak's work.

Contribution

It generalizes the finite module property to Schwartz functions and provides a decomposition of covariant distributions into invariant components with covariant polynomials.

Findings

Schwartz covariant functions form a finite module over invariant functions.

Covariant distributions can be expressed as sums of invariant distributions times covariant polynomials.

The results generalize previous work by Malgrange and Oksak.

Abstract

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