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

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

2009-02-09

arxiv.org/abs/0902.1383v1

Mathematics

Representation Theory

Scientific paper

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]).

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