References are to K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms", Contemp. Math. 165, 1994.

Let $\mathcal O$ be the ring of integers in a finite extension of $\mathbb Q_p$ and $N$ an integer prime to $p$. For $n\ge0$, denote by $\overline{\mathcal{MS}}^{\mathrm{ord}}(n,\mathcal O)$ the ordinary part of the $p$-adic completion of the direct limit of the spaces of (classical) modular symbols of weight $n+2$ and level $Np^r$ for $r\to\infty$, as defined by Kitagawa (§5.1). This space carries an action of the group $\Gamma=1+p\mathbb Z_p$.

Kitagawa proves (Thm. 5.3) that there is a canonical isomorphism $$ \overline{\mathcal{MS}}^{\mathrm{ord}}(n,\mathcal O) \cong \overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathcal O)(\kappa^n) $$ with $\kappa\colon\Gamma\rightarrow\mathcal O^\times$ the tautological inclusion and $(\kappa^n)$ meaning that the action of $\Gamma$ has been twisted by $\kappa^n$. Then he claimes (implicitly, in the proof of Thm. 5.6) that $$ \overline{\mathcal{MS}}^{\mathrm{ord}}(n,\mathcal O)[\varepsilon] \cong \overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathcal O)[\varepsilon\kappa^n] $$ for any finite order character $\varepsilon$ of $\Gamma$, where $[\cdot]$ denotes a character eigenspace.

But if we have an $\mathcal O[\![\Gamma]\!]$-module $M$ and characters $\chi$ and $\varepsilon$, then an easy calculation shows that $M(\chi)[\varepsilon]=M[\chi^{-1}\varepsilon]$, which seems to contradict the above. Since the above statement is essential for the control theory of $p$-adic modular symbols and thus for Kitagawa's construction, it is hopefully true. What is wrong here?

See also the related question Control theory for Kitagawa's $\Lambda$-adic modular symbols.


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