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Here $V$ is a representation of $G$ and $v_0$ is an $H$-fixed vector in $V$. Then the following questions naturally arise: (Q1) when is the correspondence $\\theta\\mapsto p_\\theta$ injective? (Q2) when do distinct pairs $(V,v_0)$ and $(V',v_0')$ generate the same family? In this paper, we answer these two questions (Theorems 1 and 2). 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