关于紧群作用下的协变函数和分布
On covariant functions and distributions under the action of a compact group
摘要：
设$G$是线性作用于有限维向量空间$E$上的$GL_n(\R)$的紧子群. B. Malgrange证明了$\mathcal{C}^\infty$和$G$协变函数的空间$\mathcal{C}^\infty(\R^n,E)^G$是$\mathcal{C}^\infty$和$G$不变函数的环$\mathcal{C}^\infty(\R^n)^G$上的有限模。首先，我们将这个结果推广到了$G$ -协变函数的施瓦兹空间$\mathscr{S}(\R^n,E)^G$。其次，我们证明了任何$G$ -协变分布都可以分解为$G$ -不变分布乘以一个固定的$G$ -协变多项式族的和。这是（[O]）中一个Oksak结果的推广.
英文摘要：
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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