These are two questions regarding Rubin`s paper - Global units and Ideal class groups in Inventiones 1987. The two questions are from section 3.
Let $p$ be a rational prime and $K \subset E \subset F$, where $F$ is an abelian extension of $K$ containing the Hilbert Class Field of $K$, with $G=Gal(F/K)$. Let $\rho$ be an irreducible $\mathbb{Z}_p$-representation of $\Delta = Gal(F/E)$. For any $\mathbb{Z}[\Delta]$-module $M$, write $M^{\rho}$ for $\rho$-eigenspace of $\hat{M}= \lim M/p^n M$. Explicitly, $M^{\rho}= e_{\rho}\hat{M}$, where $e_{\rho}$ is the $\rho$-idempotent. Let $W$ be a submodule of $(O_F^*)^{\rho}$ such that $(O_F^*)^{\rho}$ has no $\mathbb{Z}_p$ torsion. Let $N=p^n$.
Since $\rho \ne 1$, I have two questions:
1) Why is $(\mathbb{Z}/N\mathbb{Z})[G]^{\rho}$ contained in the augmentation ideal of $(\mathbb{Z}/N\mathbb{Z})[G]$? I couldn`t understand the action of $\rho$ on a $(\mathbb{Z}/N\mathbb{Z})[G]$-module $M$.
2) Let $V=W/W^N$. Why is $O_K^* \cap V = 0$?
